RSD characterization for n=4 agents
(automatically generated proof).

We prove that that the Random Serial Dictatorship is the only mechanism satisfying
  -- Symmetry (Anonymity + Neutrality);
  -- SD-Strategy-Proofness;
  -- Ex-Post Pareto Optimality.

Some nuances:
  -- Symmetry allows us to fix the preference of the first agent and hence to reduce the number of profiles to consider.
  -- SD-Strategy-Proofness is equivalent to the combination of 3 properties under the swap of the two adjancent objects in agent's ranking:
a) upper-invariance  b) lower invariance c) swap monotonicity.
We don't use swap monotonicity in the proof.
  -- Instead of Ex-Post Pareto Optimality we use a slightly weaker property.
Namely, if an agent does not receive a certain object in any Pareto-optimal allocation, then the mechanism must never allocate this object to this agent.

Let's start!

Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > b > c > d
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 6/24 6/24 6/24 
agent 2: 6/24 6/24 6/24 6/24 
agent 3: 6/24 6/24 6/24 6/24 
agent 4: 6/24 6/24 6/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > b > d > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 6/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 6/24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 */24 */24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > b > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 6/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 6/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 0/24 6/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 0/24 6/24 6/24 
agent 2: 4/24 8/24 6/24 6/24 
agent 3: 4/24 8/24 6/24 6/24 
agent 4: 4/24 8/24 6/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 6/24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 */24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 4/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 4/24 6/24 8/24 6/24 
agent 3: 4/24 6/24 8/24 6/24 
agent 4: 4/24 6/24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 6/24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 4/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: 2/24 8/24 8/24 6/24 
agent 4: 2/24 8/24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 2/24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: 2/24 8/24 8/24 6/24 
agent 4: 2/24 8/24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 */24 */24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > c > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 6/24 */24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 0/24 6/24 
agent 2: 6/24 4/24 8/24 6/24 
agent 3: 6/24 4/24 8/24 6/24 
agent 4: 6/24 4/24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 4/24 */24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 6/24 12/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 6/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 6/24 0/24 
agent 2: 6/24 4/24 6/24 8/24 
agent 3: 6/24 4/24 6/24 8/24 
agent 4: 6/24 4/24 6/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 4/24 6/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 6/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 2/24 8/24 6/24 8/24 
agent 3: 2/24 8/24 6/24 8/24 
agent 4: 2/24 8/24 6/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 */24 6/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 6/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 2/24 6/24 8/24 8/24 
agent 3: 2/24 6/24 8/24 8/24 
agent 4: 2/24 6/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 6/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 6/24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 6/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 18/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 18/24 0/24 0/24 
agent 2: 6/24 2/24 8/24 8/24 
agent 3: 6/24 2/24 8/24 8/24 
agent 4: 6/24 2/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 18/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 8/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 2/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > c > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 6/24 18/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 18/24 0/24 0/24 
agent 2: 6/24 2/24 8/24 8/24 
agent 3: 6/24 2/24 8/24 8/24 
agent 4: 6/24 2/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > b > d > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 6/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 */24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 2/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: */24 */24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 0/24 12/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 0/24 12/24 0/24 
agent 2: 4/24 8/24 4/24 8/24 
agent 3: 4/24 8/24 4/24 8/24 
agent 4: 4/24 8/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > b > d > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 6/24 12/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: */24 */24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 4/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > b > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 6/24 6/24 12/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 6/24 12/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 6/24 6/24 4/24 8/24 
agent 4: 6/24 6/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 4/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 */24 4/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > b > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 6/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 12/24 4/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 2/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 16/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 2/24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 16/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 2/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 5/24 8/24 
agent 3: */24 */24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 14/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 5/24 8/24 
agent 3: */24 */24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 12/24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 16/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 16/24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: */24 */24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 1/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 */24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 1/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 0/24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 24/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 24/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 24/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 24/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 */24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 0/24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 12/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 24/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 24/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 24/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 24/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 24/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 24/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 6/24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 */24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 6/24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 4/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 6/24 0/24 4/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 2/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 5/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 4/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 2/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 2/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 0/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 */24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 8/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 */24 0/24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 16/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 6/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 */24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 16/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 4/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 6/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 1/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 24/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 24/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 */24 8/24 0/24 
agent 3: 0/24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 4/24 
agent 3: 0/24 */24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 12/24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 */24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 4/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 6/24 0/24 4/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 */24 */24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 5/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 4/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 */24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 2/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 */24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 8/24 16/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 */24 16/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 24/24 0/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 */24 16/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 22/24 0/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 4/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 0/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 12/24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 22/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 0/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 1/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 22/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 24/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 24/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: */24 */24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 1/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 2/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 2/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 1/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 12/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 0/24 24/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 0/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 12/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 12/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 12/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 1/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 4/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 24/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 0/24 22/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: */24 */24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 2/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 */24 8/24 0/24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 6/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 */24 6/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 0/24 6/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 0/24 6/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 2/24 0/24 6/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 4/24 6/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 6/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 6/24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 4/24 6/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 4/24 6/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 4/24 6/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 5/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 6/24 14/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 5/24 
agent 3: */24 */24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 */24 14/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 8/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 6/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 6/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 6/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 6/24 */24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 16/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 16/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 16/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 0/24 24/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 8/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 8/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 8/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 4/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 24/24 0/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 24/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 24/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 22/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 12/24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 24/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 6/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 22/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 2/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: */24 */24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 1/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 22/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 24/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 22/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 24/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 24/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 0/24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 24/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: */24 */24 12/24 12/24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 24/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 12/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 24/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 24/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 24/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 0/24 24/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 8/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 4/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 0/24 */24 
agent 4: */24 0/24 24/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 4/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 20/24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 20/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 20/24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 20/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: */24 20/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 20/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 4/24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: */24 20/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 4/24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 4/24 0/24 
agent 3: 8/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 12/24 12/24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 24/24 0/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 0/24 */24 
agent 4: 0/24 0/24 24/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 12/24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 0/24 24/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 24/24 0/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 24/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 8/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: */24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 8/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 4/24 0/24 
agent 3: 8/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 12/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 12/24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 24/24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 12/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 4/24 0/24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 24/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 24/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 */24 22/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: */24 */24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 1/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 24/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 24/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 24/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 1/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 24/24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 24/24 0/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 */24 24/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 */24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 24/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: */24 12/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 24/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 24/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 16/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 16/24 8/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 16/24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 5/24 8/24 
agent 3: */24 */24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: */24 */24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 12/24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 24/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: */24 12/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 */24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 12/24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 12/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 12/24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 */24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 24/24 0/24 
agent 3: */24 12/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 */24 24/24 0/24 
agent 3: */24 12/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 */24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 12/24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 0/24 24/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 0/24 24/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 0/24 24/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 6/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 6/24 0/24 16/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 24/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 16/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 12/24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 1/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 14/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 22/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 5/24 
agent 3: */24 */24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 6/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 6/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 8/24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 6/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 2/24 6/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 6/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 2/24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 6/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 0/24 6/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 2/24 0/24 6/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 2/24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 */24 4/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 24/24 0/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 4/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 */24 */24 
agent 2: 0/24 */24 24/24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 24/24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 4/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > c > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 */24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 24/24 0/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 8/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 4/24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 8/24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 8/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 8/24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 4/24 0/24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 8/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 16/24 */24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: 8/24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 */24 */24 
agent 2: 0/24 */24 24/24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 12/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 12/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 24/24 0/24 
agent 3: */24 12/24 0/24 12/24 
agent 4: */24 12/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 */24 */24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 */24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 16/24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 16/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 0/24 4/24 */24 8/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 4/24 
agent 3: 0/24 */24 12/24 8/24 
agent 4: 0/24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 8/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 2/24 */24 0/24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 */24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 4/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 4/24 0/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: 0/24 8/24 8/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 12/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 16/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: */24 0/24 16/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 20/24 */24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 */24 */24 
agent 2: */24 0/24 24/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 */24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 12/24 0/24 12/24 
agent 4: */24 12/24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 12/24 12/24 
agent 4: */24 */24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 12/24 0/24 12/24 
agent 4: 0/24 12/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 20/24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 8/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 4/24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 4/24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 4/24 
agent 3: 0/24 */24 12/24 8/24 
agent 4: 0/24 */24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 20/24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 4/24 
agent 3: 0/24 */24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 20/24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 12/24 
agent 4: 0/24 */24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: 0/24 8/24 */24 4/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 */24 16/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: */24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 0/24 24/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 4/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 4/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 4/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: */24 */24 0/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 4/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: */24 */24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 16/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 1/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 1/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: */24 */24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 12/24 0/24 12/24 
agent 4: 0/24 12/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: */24 */24 12/24 12/24 
agent 4: */24 */24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 */24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 */24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 12/24 */24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: */24 */24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 4/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 */24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 8/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 8/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 4/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 12/24 
agent 3: 8/24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 4/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 1/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 1/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 1/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 1/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 12/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 12/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 0/24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 16/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: */24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: */24 */24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: */24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 4/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 4/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: */24 */24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 0/24 
agent 2: 8/24 8/24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: */24 */24 */24 1/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: */24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 4/24 */24 
agent 4: */24 */24 */24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 1/24 12/24 
agent 3: */24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: 4/24 */24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 12/24 
agent 4: */24 */24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 16/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 16/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 */24 
agent 4: */24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 16/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 */24 
agent 4: */24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 24/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 20/24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 20/24 */24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 20/24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 1/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 */24 4/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 4/24 */24 
agent 4: */24 */24 */24 5/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 */24 */24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 16/24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 0/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 0/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 */24 16/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 */24 16/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 4/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 20/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 4/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 8/24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 0/24 8/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 0/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 8/24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 0/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 0/24 8/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 8/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 12/24 12/24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 8/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 4/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 8/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 */24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 8/24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 12/24 */24 0/24 
agent 3: 8/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 12/24 */24 0/24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 12/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 16/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 0/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 0/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: 8/24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 24/24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 12/24 */24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 24/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 */24 */24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 4/24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: */24 */24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 0/24 
agent 2: 8/24 8/24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: */24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: */24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 2/24 
agent 3: 8/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 0/24 
agent 2: 8/24 */24 */24 2/24 
agent 3: 8/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 8/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 12/24 */24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 20/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 20/24 */24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 */24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 20/24 */24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 4/24 20/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 4/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 4/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 20/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 8/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 8/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 0/24 
agent 2: 8/24 8/24 */24 2/24 
agent 3: 8/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 */24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 12/24 
agent 3: 8/24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 4/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 4/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 16/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 16/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 12/24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 12/24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 12/24 */24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 22/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 16/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 20/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 0/24 24/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 8/24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 12/24 12/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 20/24 4/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 */24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: */24 */24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 */24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 16/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 1/24 12/24 
agent 3: */24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 1/24 12/24 
agent 3: */24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 1/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 1/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: */24 */24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 0/24 
agent 2: 8/24 8/24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 8/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 0/24 
agent 2: 8/24 8/24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: 4/24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: 4/24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 4/24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: */24 4/24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 4/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 12/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 2/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 2/24 
agent 3: 0/24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 2/24 
agent 3: 0/24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 2/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 2/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 0/24 
agent 2: 8/24 8/24 */24 2/24 
agent 3: 8/24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 4/24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: */24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: */24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 0/24 
agent 2: 8/24 8/24 */24 2/24 
agent 3: 8/24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 4/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 0/24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 0/24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 8/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: */24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 */24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: */24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 8/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 20/24 */24 */24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 20/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 20/24 */24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 20/24 */24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 0/24 */24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 4/24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 0/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 0/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 4/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 8/24 */24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 8/24 */24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 4/24 6/24 14/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: 8/24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 8/24 5/24 
agent 3: */24 */24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: 8/24 */24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 8/24 5/24 
agent 3: */24 */24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 8/24 5/24 
agent 3: */24 */24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 */24 7/24 */24 
agent 4: 0/24 */24 4/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 13/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 7/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 6/24 2/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 */24 1/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 1/24 12/24 
agent 3: */24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 6/24 2/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 6/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 */24 6/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 6/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 6/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 4/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 4/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: */24 */24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 0/24 
agent 2: 8/24 8/24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 2/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: */24 16/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 0/24 
agent 3: */24 20/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 */24 0/24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: */24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: */24 */24 */24 0/24 
agent 4: */24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 2/24 */24 0/24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: */24 16/24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 20/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 20/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 12/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 12/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 0/24 24/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 1/24 12/24 
agent 3: */24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: */24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 20/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 20/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > d > a
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 8/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 20/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 */24 12/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 20/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 */24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 20/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: */24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 12/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: */24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 12/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 12/24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 */24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 6/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 6/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 4/24 6/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 4/24 6/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 5/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 6/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 4/24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 5/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 11/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 4/24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 4/24 6/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 4/24 6/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 5/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 6/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 5/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 11/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 4/24 6/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 11/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 6/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 10/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 10/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 10/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 14/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 5/24 8/24 
agent 3: */24 */24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 5/24 8/24 
agent 3: */24 */24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 5/24 8/24 
agent 3: */24 */24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 3/24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 5/24 8/24 
agent 3: 8/24 3/24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 13/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 3/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 3/24 */24 21/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 3/24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 21/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 2/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 2/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 2/24 */24 0/24 16/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 2/24 */24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 2/24 6/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 2/24 6/24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 2/24 6/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 8/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 12/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 12/24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 12/24 6/24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 6/24 12/24 6/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 0/24 4/24 12/24 
agent 4: 0/24 */24 6/24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 6/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 8/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 4/24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 8/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 */24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 8/24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 8/24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: 0/24 8/24 */24 4/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 8/24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 8/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: 0/24 8/24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 */24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 8/24 12/24 */24 
agent 4: 0/24 8/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > c > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 8/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 6/24 0/24 8/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 10/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 6/24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > c > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 */24 
agent 4: 0/24 8/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: 0/24 8/24 8/24 */24 
agent 4: 0/24 8/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 8/24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 8/24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 8/24 
agent 3: 0/24 8/24 */24 8/24 
agent 4: 0/24 8/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 8/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 6/24 0/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 6/24 0/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 6/24 0/24 10/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 6/24 0/24 10/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 10/24 14/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 8/24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 8/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 8/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: 0/24 */24 8/24 8/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 8/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: 0/24 */24 8/24 8/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 */24 8/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 8/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 8/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 8/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 8/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 8/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 8/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 */24 8/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 16/24 */24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 */24 8/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 8/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 8/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 8/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 8/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 4/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 8/24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 10/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: 0/24 0/24 8/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 10/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 0/24 */24 5/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 0/24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 14/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 16/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 0/24 16/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 14/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 14/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 4/24 14/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 5/24 8/24 
agent 3: */24 */24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 5/24 8/24 
agent 3: */24 */24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 5/24 8/24 
agent 3: */24 */24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: */24 */24 */24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 4/24 */24 
agent 4: */24 */24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 7/24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 3/24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 5/24 8/24 
agent 3: 8/24 3/24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 13/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 4/24 */24 
agent 4: 0/24 */24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 3/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 2/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 3/24 0/24 21/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 2/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 6/24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 3/24 0/24 21/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 */24 1/24 20/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 2/24 6/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 2/24 6/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 4/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 6/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 2/24 0/24 6/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 2/24 0/24 6/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 2/24 0/24 6/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 4/24 6/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 2/24 */24 6/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 4/24 6/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 5/24 
agent 3: 8/24 */24 6/24 5/24 
agent 4: 0/24 4/24 6/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 5/24 
agent 3: 8/24 5/24 6/24 5/24 
agent 4: 0/24 4/24 6/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 5/24 6/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 2/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 6/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 2/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 6/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 4/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 6/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 6/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 6/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 6/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: 8/24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: 8/24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 1/24 12/24 
agent 3: */24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 */24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 1/24 12/24 
agent 3: */24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 1/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 4/24 */24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 4/24 */24 
agent 4: 0/24 */24 */24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 */24 
agent 4: 0/24 */24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 */24 7/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 8/24 */24 4/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: */24 */24 */24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 6/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 6/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 6/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 6/24 6/24 
agent 2: 8/24 4/24 6/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 12/24 6/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: */24 */24 6/24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 4/24 14/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 8/24 */24 4/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: */24 */24 6/24 6/24 
agent 3: */24 */24 6/24 6/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 4/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 12/24 */24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > c > d
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: */24 */24 6/24 6/24 
agent 3: */24 */24 6/24 6/24 
agent 4: */24 */24 6/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 6/24 
agent 2: */24 6/24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 6/24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 6/24 */24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: 2/24 */24 0/24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 6/24 
agent 4: 0/24 4/24 14/24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 6/24 
agent 2: 2/24 */24 0/24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 2/24 0/24 16/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: */24 16/24 0/24 6/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 */24 */24 6/24 
agent 4: 2/24 0/24 16/24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 */24 */24 6/24 
agent 4: 2/24 0/24 16/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 2/24 0/24 16/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 6/24 
agent 4: 0/24 4/24 14/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 2/24 0/24 6/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 4/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 16/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 6/24 
agent 4: 0/24 4/24 14/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 6/24 
agent 4: 0/24 4/24 14/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 0/24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 5/24 5/24 6/24 
agent 4: 0/24 4/24 14/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 4/24 6/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 6/24 */24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: */24 */24 */24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 14/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 4/24 6/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 4/24 6/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 5/24 6/24 5/24 
agent 4: 0/24 4/24 6/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 2/24 0/24 6/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 3/24 8/24 5/24 
agent 3: */24 */24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 5/24 6/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 0/24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 2/24 0/24 6/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 6/24 */24 12/24 
agent 4: */24 */24 4/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 6/24 */24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 8/24 */24 12/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 6/24 */24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 8/24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 8/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: */24 */24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 4/24 6/24 6/24 8/24 
agent 3: 4/24 6/24 6/24 8/24 
agent 4: 4/24 6/24 6/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: */24 */24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 6/24 6/24 */24 
agent 3: */24 5/24 6/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 6/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 4/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 2/24 8/24 6/24 8/24 
agent 3: 2/24 8/24 6/24 8/24 
agent 4: 2/24 8/24 6/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: 2/24 */24 6/24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > a > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 2/24 6/24 8/24 8/24 
agent 3: 2/24 6/24 8/24 8/24 
agent 4: 2/24 6/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 8/24 
agent 4: 0/24 */24 8/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > c > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 */24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 */24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 16/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 16/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 */24 4/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 4/24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 */24 4/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 */24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 */24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 */24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 18/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 4/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 18/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 4/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 4/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 6/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 8/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 */24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 8/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: */24 */24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 */24 7/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 12/24 
agent 3: */24 0/24 4/24 12/24 
agent 4: 0/24 */24 6/24 0/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 12/24 
agent 3: */24 0/24 4/24 12/24 
agent 4: 0/24 */24 6/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 0/24 4/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 18/24 6/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 4/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 18/24 6/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 4/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 18/24 */24 0/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: */24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: */24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 0/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 0/24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 6/24 */24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 6/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: 8/24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 12/24 0/24 
agent 3: 8/24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 4/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 4/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 18/24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 4/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 18/24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 4/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 18/24 6/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 4/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 18/24 6/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 4/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 18/24 6/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 11/24 1/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 */24 12/24 
agent 4: 0/24 */24 1/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 */24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: */24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 */24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 24/24 0/24 0/24 
agent 3: */24 0/24 24/24 */24 
agent 4: */24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 */24 
agent 4: 0/24 0/24 */24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 16/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 16/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 */24 
agent 4: */24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 16/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 */24 
agent 4: */24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: */24 0/24 16/24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 12/24 4/24 */24 
agent 4: */24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: */24 0/24 16/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 2/24 0/24 16/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 2/24 0/24 16/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 6/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 2/24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: */24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: */24 0/24 4/24 12/24 
agent 4: */24 */24 */24 0/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: */24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 8/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 18/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 18/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 8/24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 8/24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 18/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 18/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 6/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 6/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 2/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 0/24 8/24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 0/24 8/24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: */24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: */24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 */24 2/24 
agent 3: */24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: */24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: */24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 4/24 6/24 12/24 */24 
agent 4: 0/24 4/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: */24 */24 12/24 7/24 
agent 4: */24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 2/24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: */24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 4/24 8/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 2/24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: */24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 3/24 0/24 21/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 6/24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 3/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 3/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 6/24 2/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 3/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 6/24 2/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 6/24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 6/24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 4/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 2/24 6/24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 16/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: */24 0/24 */24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 4/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: */24 0/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 */24 */24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: */24 0/24 16/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 */24 
agent 4: */24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 */24 
agent 4: */24 0/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 11/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 */24 
agent 4: */24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: 2/24 */24 6/24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: 2/24 */24 6/24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 6/24 6/24 */24 
agent 3: */24 5/24 6/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 16/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 6/24 6/24 */24 
agent 3: */24 5/24 6/24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 6/24 6/24 */24 
agent 3: */24 5/24 6/24 */24 
agent 4: */24 */24 6/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 5/24 6/24 */24 
agent 4: 0/24 */24 6/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 16/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 16/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: 2/24 12/24 6/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 6/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 6/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 6/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 4/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 4/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 4/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 8/24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 16/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 8/24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 4/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 4/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 4/24 0/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 12/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 8/24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 6/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 6/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 0/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 0/24 8/24 
agent 3: */24 */24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 */24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 8/24 
agent 3: */24 */24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 */24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 3/24 9/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 6/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 6/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 */24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 */24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 3/24 9/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 */24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 3/24 */24 0/24 
agent 3: */24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 7/24 12/24 
agent 4: */24 3/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 8/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 8/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 18/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 6/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 18/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: */24 */24 2/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 */24 1/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 18/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 18/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 6/24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: */24 6/24 */24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 12/24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 12/24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 3/24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 */24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 */24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 0/24 
agent 2: 8/24 8/24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 1/24 12/24 
agent 3: */24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 1/24 12/24 
agent 3: */24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 1/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: */24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: */24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 0/24 
agent 2: 8/24 8/24 */24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: */24 */24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 21/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 6/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 4/24 8/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 4/24 */24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 8/24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 6/24 */24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 6/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 6/24 10/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 6/24 10/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: 0/24 */24 8/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 10/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: 0/24 */24 8/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 3/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 3/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 3/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 */24 
agent 4: */24 */24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 6/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 6/24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 */24 8/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 */24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 */24 
agent 4: */24 */24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 4/24 6/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 4/24 6/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 */24 0/24 
agent 3: */24 */24 7/24 12/24 
agent 4: */24 3/24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 6/24 2/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 6/24 2/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 4/24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 4/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: */24 4/24 8/24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 */24 10/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 5/24 
agent 3: */24 3/24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 10/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 5/24 
agent 3: */24 3/24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 5/24 
agent 3: 8/24 3/24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 8/24 5/24 
agent 3: 8/24 3/24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 */24 
agent 4: 0/24 2/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 6/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 8/24 */24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 4/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 4/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 10/24 4/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 10/24 4/24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 10/24 4/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 10/24 4/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 2/24 10/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 10/24 4/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 10/24 4/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 10/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 10/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 8/24 
agent 3: 8/24 */24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 8/24 
agent 3: 8/24 */24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 8/24 
agent 3: 8/24 */24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 3/24 5/24 8/24 
agent 3: 8/24 3/24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 12/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 3/24 5/24 8/24 
agent 3: 8/24 3/24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 5/24 8/24 
agent 3: 8/24 3/24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 8/24 5/24 
agent 3: 8/24 3/24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 6/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 6/24 6/24 
agent 2: 8/24 4/24 6/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 12/24 6/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 14/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 0/24 
agent 2: 8/24 */24 */24 8/24 
agent 3: 8/24 */24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 6/24 6/24 
agent 4: */24 */24 */24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 4/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 12/24 6/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 2/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 2/24 6/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 */24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 0/24 
agent 2: 8/24 */24 */24 8/24 
agent 3: 8/24 */24 4/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 */24 0/24 
agent 2: 8/24 */24 */24 8/24 
agent 3: 8/24 */24 4/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 12/24 0/24 
agent 2: 8/24 */24 */24 8/24 
agent 3: 8/24 4/24 4/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 12/24 0/24 
agent 2: 8/24 4/24 4/24 8/24 
agent 3: 8/24 4/24 4/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 */24 7/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 */24 7/24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 4/24 7/24 
agent 4: */24 */24 4/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 5/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 5/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 */24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 16/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 16/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 16/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 16/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 6/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 6/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 6/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 6/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 4/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 16/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 */24 6/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 6/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 2/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 2/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 2/24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 2/24 4/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 2/24 12/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: */24 16/24 0/24 6/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 2/24 12/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: */24 12/24 4/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: */24 16/24 0/24 6/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 16/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 4/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 18/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 6/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 6/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 6/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 6/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 6/24 0/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 16/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 6/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 6/24 0/24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 14/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 */24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 1/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 */24 2/24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 */24 2/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 14/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 14/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 8/24 */24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 */24 8/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 4/24 0/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 4/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 0/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 0/24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 0/24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 11/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 0/24 
agent 2: 8/24 */24 */24 8/24 
agent 3: 8/24 */24 4/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 */24 0/24 
agent 2: 8/24 */24 */24 8/24 
agent 3: 8/24 */24 4/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 12/24 0/24 
agent 2: 8/24 */24 */24 8/24 
agent 3: 8/24 4/24 4/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 12/24 0/24 
agent 2: 8/24 4/24 4/24 8/24 
agent 3: 8/24 4/24 4/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 12/24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 2/24 10/24 4/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 3/24 5/24 8/24 
agent 3: 8/24 3/24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 12/24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 2/24 10/24 4/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 3/24 5/24 8/24 
agent 3: 8/24 3/24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 5/24 8/24 
agent 3: 8/24 3/24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 8/24 5/24 
agent 3: */24 */24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 8/24 5/24 
agent 3: */24 */24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 8/24 5/24 
agent 3: 8/24 3/24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 8/24 4/24 
agent 2: 8/24 4/24 8/24 4/24 
agent 3: 8/24 4/24 8/24 4/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 4/24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 4/24 
agent 4: */24 */24 0/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > b > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 6/24 0/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 8/24 */24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 */24 8/24 */24 
agent 4: 0/24 12/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 4/24 8/24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 */24 */24 4/24 
agent 4: */24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 4/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 4/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 4/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 12/24 4/24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 4/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 12/24 4/24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 4/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 4/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 16/24 */24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 5/24 5/24 
agent 3: 8/24 */24 5/24 5/24 
agent 4: 0/24 0/24 10/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 5/24 5/24 
agent 3: 8/24 6/24 5/24 5/24 
agent 4: 0/24 0/24 10/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: */24 0/24 11/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: */24 0/24 11/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 6/24 0/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 7/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 7/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 8/24 */24 5/24 5/24 
agent 3: 8/24 6/24 5/24 5/24 
agent 4: 0/24 0/24 10/24 14/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 7/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 */24 17/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 3/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 3/24 12/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 4/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 */24 
agent 3: */24 2/24 12/24 6/24 
agent 4: 4/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 */24 
agent 3: */24 2/24 12/24 6/24 
agent 4: 4/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: */24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 21/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: */24 2/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 6/24 0/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 2/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 2/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 2/24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 0/24 
agent 3: 2/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 2/24 */24 6/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 2/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 6/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 2/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 */24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 */24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 16/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 3/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 3/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: */24 0/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 6/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 6/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 12/24 12/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 6/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: */24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 6/24 0/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 2/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 */24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 16/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 16/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 16/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: */24 2/24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: */24 2/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 2/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 0/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 22/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 4/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 2/24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 */24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 12/24 2/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 6/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 6/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 6/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 6/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 6/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 6/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 6/24 18/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 */24 6/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 5/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 7/24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 14/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 14/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 2/24 */24 6/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 2/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 6/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 4/24 2/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 2/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 4/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 */24 */24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 */24 */24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 8/24 */24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 */24 */24 
agent 4: */24 */24 2/24 10/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 8/24 */24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 8/24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 8/24 4/24 
agent 4: */24 */24 8/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 4/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 12/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 2/24 6/24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 6/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 6/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 2/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 2/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 2/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: */24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 6/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 6/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 6/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 6/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 22/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 22/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 1/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: */24 */24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 8/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 24/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 0/24 12/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 24/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 */24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 12/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 3/24 12/24 9/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 3/24 12/24 9/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 3/24 0/24 21/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 3/24 0/24 21/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 2/24 6/24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 */24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 2/24 6/24 12/24 */24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 1/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 1/24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 1/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 1/24 14/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 13/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 2/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 13/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 2/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 2/24 4/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 */24 0/24 
agent 3: */24 3/24 7/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 */24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 2/24 4/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 2/24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 */24 0/24 
agent 3: */24 3/24 7/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: */24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: */24 3/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 0/24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 1/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 1/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 1/24 5/24 10/24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 1/24 */24 10/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 12/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: */24 8/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 12/24 6/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 2/24 12/24 6/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 4/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 16/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 2/24 12/24 6/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 */24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 6/24 18/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 8/24 14/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 0/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 0/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 11/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 12/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 6/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 6/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 6/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 6/24 6/24 
agent 2: 8/24 4/24 6/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 12/24 6/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 6/24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 6/24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 4/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 12/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 6/24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 0/24 6/24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 0/24 6/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 6/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 2/24 0/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 4/24 6/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 6/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 2/24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 2/24 0/24 16/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 2/24 0/24 16/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 2/24 0/24 */24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 2/24 4/24 12/24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: */24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 0/24 6/24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 6/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 18/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 2/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 6/24 12/24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 2/24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 2/24 */24 
agent 2: 12/24 0/24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 2/24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 8/24 
agent 4: */24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 8/24 
agent 4: */24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 */24 6/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: */24 */24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 12/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 16/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 */24 
agent 3: 8/24 0/24 */24 16/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 20/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: 0/24 4/24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: 0/24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 */24 */24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 4/24 
agent 2: */24 */24 */24 4/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 11/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 4/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 4/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 18/24 0/24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 10/24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 14/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 14/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 */24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 */24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: */24 */24 2/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 18/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 18/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 18/24 0/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 18/24 0/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 0/24 
agent 2: 8/24 0/24 4/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 18/24 6/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 10/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 0/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 0/24 
agent 2: 8/24 0/24 4/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 18/24 6/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 4/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 18/24 6/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 11/24 1/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 */24 12/24 
agent 4: 0/24 */24 1/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: */24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 3/24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 2/24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 2/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 2/24 */24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 */24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 21/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 */24 0/24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 4/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 */24 4/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: 0/24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 16/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 */24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 4/24 12/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 6/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 8/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 */24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 3/24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 */24 1/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 18/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 4/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 18/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 18/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 18/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 6/24 */24 8/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 6/24 */24 8/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 6/24 10/24 8/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 10/24 8/24 
agent 4: 0/24 6/24 10/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 10/24 8/24 
agent 4: 0/24 6/24 10/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 10/24 8/24 
agent 4: 0/24 6/24 10/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 3/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: */24 */24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: */24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 21/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: */24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: */24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 */24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 */24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 2/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 2/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 */24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 6/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 9/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 */24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 1/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 8/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 8/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 2/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 22/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 22/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 22/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 22/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 22/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 */24 0/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 */24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 6/24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 6/24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 */24 12/24 2/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 8/24 */24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 */24 
agent 4: 0/24 0/24 6/24 18/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 8/24 */24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 2/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 2/24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 2/24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 2/24 12/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 2/24 12/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 12/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 12/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 */24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 0/24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 0/24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 10/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 10/24 0/24 */24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 10/24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 10/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 3/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 */24 3/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 */24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 9/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 9/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 2/24 */24 */24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 9/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 9/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: */24 3/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 3/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 1/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 3/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: 8/24 */24 3/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 1/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 3/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 3/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: 0/24 9/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: 0/24 9/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 3/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 0/24 
agent 2: 8/24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 */24 0/24 
agent 2: 8/24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 12/24 0/24 
agent 2: 8/24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 12/24 0/24 
agent 2: 8/24 4/24 4/24 8/24 
agent 3: 8/24 4/24 4/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 */24 7/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 5/24 8/24 
agent 3: */24 */24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 4/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 */24 7/24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 4/24 7/24 
agent 4: */24 */24 4/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: */24 */24 5/24 8/24 
agent 3: */24 */24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: */24 */24 5/24 8/24 
agent 3: */24 */24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 5/24 8/24 
agent 3: 8/24 3/24 5/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 8/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 3/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 8/24 5/24 
agent 3: 8/24 3/24 8/24 5/24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 */24 10/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 8/24 3/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 */24 10/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 */24 10/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 12/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 6/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 6/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 6/24 2/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 */24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 */24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 */24 1/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 */24 1/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 3/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 */24 1/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 */24 4/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 21/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 */24 1/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 6/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 */24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 */24 18/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 1/24 5/24 10/24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 1/24 5/24 10/24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 1/24 5/24 10/24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 6/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 6/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 18/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 4/24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 2/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 3/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 3/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 21/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 21/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 21/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 */24 0/24 18/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 3/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 21/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 */24 0/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 6/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 6/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 6/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 6/24 8/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 */24 6/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: 0/24 4/24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 6/24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 8/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 */24 6/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 6/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 6/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 7/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 */24 4/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 6/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 6/24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 8/24 */24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 7/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 8/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 */24 2/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 4/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 8/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 4/24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 8/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 8/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 8/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 8/24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 8/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 8/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 8/24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 4/24 
agent 3: */24 8/24 0/24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 4/24 
agent 3: */24 8/24 0/24 */24 
agent 4: 0/24 8/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 4/24 
agent 3: */24 8/24 0/24 */24 
agent 4: 0/24 8/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 8/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 8/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 2/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 0/24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 8/24 */24 16/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 0/24 */24 
agent 4: 0/24 8/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 0/24 */24 
agent 4: 0/24 8/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 4/24 8/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 10/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 10/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 6/24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 10/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 6/24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 6/24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 8/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 6/24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 8/24 */24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 6/24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 8/24 */24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 8/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 8/24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 8/24 */24 4/24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 8/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: */24 8/24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 8/24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 8/24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: */24 8/24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 8/24 */24 4/24 
agent 4: 0/24 8/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 8/24 */24 4/24 
agent 4: 0/24 8/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 4/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 8/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: */24 12/24 */24 6/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: */24 12/24 */24 6/24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: */24 12/24 */24 6/24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 10/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 10/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 10/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 4/24 8/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 4/24 8/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 4/24 8/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 8/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 8/24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 8/24 2/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 8/24 4/24 
agent 4: */24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 8/24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 8/24 4/24 
agent 4: */24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 8/24 4/24 
agent 4: */24 */24 8/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 */24 6/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 6/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 6/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 10/24 
agent 4: 0/24 8/24 6/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 10/24 
agent 4: 0/24 8/24 6/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 4/24 
agent 3: */24 8/24 */24 10/24 
agent 4: 0/24 8/24 6/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 4/24 
agent 3: 0/24 */24 4/24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 10/24 
agent 4: 0/24 8/24 6/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 10/24 
agent 4: 0/24 8/24 6/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 6/24 10/24 
agent 4: 0/24 8/24 6/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 4/24 
agent 3: 0/24 8/24 6/24 10/24 
agent 4: 0/24 8/24 6/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 10/24 
agent 4: 0/24 */24 4/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 10/24 
agent 4: 0/24 10/24 4/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 4/24 
agent 3: 0/24 */24 4/24 10/24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 */24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 */24 4/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 10/24 4/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 10/24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 10/24 */24 10/24 
agent 4: 0/24 10/24 4/24 10/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 10/24 4/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 10/24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: 2/24 10/24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 */24 */24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 10/24 */24 10/24 
agent 4: 0/24 10/24 4/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 10/24 4/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 10/24 4/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 6/24 4/24 
agent 3: 0/24 8/24 6/24 10/24 
agent 4: 0/24 8/24 6/24 10/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 2/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 6/24 4/24 
agent 3: 0/24 8/24 6/24 10/24 
agent 4: 0/24 8/24 6/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 6/24 8/24 6/24 4/24 
agent 3: 0/24 8/24 6/24 10/24 
agent 4: 0/24 8/24 6/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 6/24 8/24 6/24 4/24 
agent 3: 0/24 8/24 6/24 10/24 
agent 4: 0/24 8/24 6/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 6/24 0/24 
agent 2: 6/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 6/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 6/24 8/24 */24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 6/24 0/24 
agent 2: 6/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 6/24 0/24 
agent 2: 6/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 6/24 0/24 
agent 2: 6/24 12/24 */24 */24 
agent 3: 6/24 0/24 */24 */24 
agent 4: 6/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 6/24 0/24 */24 */24 
agent 4: */24 0/24 1/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 6/24 */24 */24 */24 
agent 4: */24 0/24 7/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 6/24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 1/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 6/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 6/24 0/24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: 6/24 */24 12/24 */24 
agent 3: 6/24 */24 12/24 */24 
agent 4: 6/24 0/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 6/24 */24 12/24 */24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 6/24 0/24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 6/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 8/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 */24 
agent 4: */24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 */24 
agent 4: */24 0/24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 8/24 2/24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 2/24 
agent 2: 6/24 */24 12/24 */24 
agent 3: 6/24 */24 12/24 */24 
agent 4: 6/24 0/24 0/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 8/24 2/24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 8/24 2/24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 8/24 2/24 
agent 2: 5/24 8/24 8/24 3/24 
agent 3: 5/24 8/24 8/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 2/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 3/24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 5/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 0/24 2/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 8/24 0/24 2/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 8/24 0/24 2/24 
agent 2: 5/24 8/24 8/24 3/24 
agent 3: 5/24 8/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 2/24 
agent 2: */24 12/24 8/24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 8/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 8/24 */24 3/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 3/24 
agent 4: 0/24 0/24 8/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 5/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 2/24 
agent 2: */24 12/24 8/24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 12/24 8/24 */24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 1/24 12/24 8/24 3/24 
agent 3: 1/24 12/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 8/24 12/24 */24 
agent 3: */24 8/24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 8/24 12/24 */24 
agent 3: */24 8/24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 1/24 8/24 12/24 3/24 
agent 3: 1/24 8/24 12/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 8/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 3/24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 8/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 8/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 1/24 8/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 2/24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 */24 8/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 3/24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 2/24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 16/24 2/24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 16/24 2/24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 4/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 16/24 0/24 2/24 
agent 2: 6/24 */24 12/24 */24 
agent 3: 6/24 */24 12/24 */24 
agent 4: 6/24 0/24 0/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: 8/24 */24 3/24 */24 
agent 4: 0/24 4/24 2/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 16/24 0/24 2/24 
agent 2: 6/24 */24 12/24 */24 
agent 3: 6/24 */24 12/24 */24 
agent 4: 6/24 0/24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 16/24 0/24 2/24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: 6/24 4/24 12/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 */24 2/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 2/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 18/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 18/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 18/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 3/24 12/24 7/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 18/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 3/24 12/24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 2/24 3/24 12/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 2/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 18/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 3/24 
agent 4: 2/24 3/24 12/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 3/24 
agent 4: 2/24 3/24 12/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 12/24 3/24 
agent 4: 2/24 3/24 12/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 18/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 7/24 12/24 3/24 
agent 4: 2/24 3/24 12/24 7/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 18/24 4/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 0/24 */24 
agent 4: 0/24 2/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 12/24 
agent 3: */24 7/24 12/24 3/24 
agent 4: 2/24 3/24 12/24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 12/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 2/24 3/24 12/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: 2/24 */24 12/24 7/24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: 2/24 */24 12/24 7/24 
agent 4: */24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: */24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 */24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: */24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 0/24 */24 
agent 4: 0/24 2/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 6/24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 1/24 5/24 */24 
agent 4: 0/24 1/24 14/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 6/24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 18/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 3/24 12/24 7/24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 18/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 3/24 12/24 7/24 
agent 4: 0/24 3/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 8/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 8/24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: */24 3/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: */24 3/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 14/24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: */24 3/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: */24 3/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 4/24 7/24 
agent 4: */24 */24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 */24 0/24 
agent 3: */24 */24 7/24 12/24 
agent 4: */24 3/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 4/24 7/24 
agent 4: */24 */24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 4/24 7/24 
agent 4: */24 */24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 11/24 4/24 7/24 
agent 4: */24 11/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: */24 11/24 4/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 6/24 6/24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 4/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 6/24 6/24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 */24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 6/24 6/24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 11/24 */24 6/24 
agent 4: */24 11/24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 */24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 11/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 */24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 7/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 7/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 4/24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 4/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 */24 4/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 */24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 4/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 */24 0/24 
agent 3: */24 */24 7/24 12/24 
agent 4: */24 3/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 */24 0/24 
agent 3: */24 3/24 7/24 12/24 
agent 4: */24 3/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: */24 3/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 8/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 */24 3/24 12/24 
agent 4: */24 3/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 8/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 8/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: */24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 4/24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: */24 4/24 8/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: */24 4/24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: */24 4/24 8/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 4/24 4/24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: 4/24 4/24 8/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 4/24 4/24 8/24 8/24 
agent 3: 4/24 4/24 8/24 8/24 
agent 4: 4/24 4/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: */24 */24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 4/24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 2/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: */24 */24 8/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 4/24 4/24 8/24 8/24 
agent 3: 4/24 4/24 8/24 8/24 
agent 4: 4/24 4/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 4/24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 8/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: */24 3/24 8/24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 8/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: */24 3/24 8/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 8/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 2/24 3/24 8/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 14/24 0/24 0/24 
agent 2: */24 4/24 8/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 2/24 3/24 8/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 8/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 2/24 3/24 8/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 6/24 0/24 
agent 2: */24 6/24 6/24 */24 
agent 3: */24 5/24 6/24 */24 
agent 4: */24 */24 6/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 */24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: */24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 */24 */24 11/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 6/24 0/24 
agent 2: */24 6/24 6/24 */24 
agent 3: */24 5/24 6/24 */24 
agent 4: */24 */24 6/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: */24 6/24 6/24 */24 
agent 3: */24 5/24 6/24 */24 
agent 4: */24 */24 6/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: */24 6/24 6/24 */24 
agent 3: */24 5/24 6/24 */24 
agent 4: */24 5/24 6/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: */24 6/24 */24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: */24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: */24 6/24 */24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: */24 */24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: */24 6/24 */24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: */24 5/24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 6/24 6/24 
agent 2: */24 */24 6/24 6/24 
agent 3: */24 */24 6/24 6/24 
agent 4: */24 */24 6/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 6/24 6/24 
agent 2: */24 */24 6/24 6/24 
agent 3: */24 */24 6/24 6/24 
agent 4: */24 */24 6/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 6/24 6/24 
agent 2: */24 */24 6/24 6/24 
agent 3: */24 */24 6/24 6/24 
agent 4: */24 */24 6/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 2/24 10/24 4/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 */24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: */24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 */24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: */24 3/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 */24 3/24 12/24 
agent 4: */24 3/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: */24 3/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 2/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: */24 3/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: */24 3/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: */24 */24 0/24 12/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 2/24 3/24 12/24 7/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 2/24 */24 */24 
agent 3: 2/24 */24 */24 11/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: */24 */24 0/24 12/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 2/24 3/24 12/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 2/24 3/24 12/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 2/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 10/24 2/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 10/24 */24 0/24 12/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 2/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 */24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 10/24 2/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 2/24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 4/24 0/24 
agent 2: 8/24 */24 5/24 5/24 
agent 3: 8/24 6/24 5/24 5/24 
agent 4: 0/24 0/24 10/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 4/24 0/24 
agent 2: 8/24 */24 5/24 5/24 
agent 3: 8/24 6/24 5/24 5/24 
agent 4: 0/24 0/24 10/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 6/24 5/24 5/24 
agent 3: 8/24 6/24 5/24 5/24 
agent 4: 0/24 0/24 10/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: */24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 6/24 5/24 5/24 
agent 3: 8/24 6/24 5/24 5/24 
agent 4: 0/24 0/24 14/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 6/24 */24 5/24 
agent 4: */24 0/24 */24 11/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 */24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 2/24 4/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 3/24 */24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: 8/24 */24 3/24 */24 
agent 4: 0/24 4/24 2/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 2/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 2/24 4/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 2/24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: */24 2/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: */24 */24 4/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 2/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 2/24 4/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 2/24 4/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 */24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 */24 
agent 4: 0/24 2/24 */24 18/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 */24 
agent 4: 0/24 2/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 2/24 4/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 7/24 12/24 
agent 4: 0/24 */24 9/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 */24 
agent 4: 0/24 */24 0/24 23/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 2/24 18/24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 7/24 12/24 
agent 4: 0/24 */24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 3/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 10/24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 */24 
agent 4: 0/24 2/24 4/24 18/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 10/24 0/24 
agent 3: 2/24 */24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 */24 
agent 4: 0/24 2/24 4/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 13/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 3/24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 */24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 3/24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 3/24 */24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 3/24 */24 
agent 4: 0/24 */24 */24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 8/24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 3/24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 3/24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 21/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 */24 0/24 18/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 21/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 */24 0/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 6/24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 6/24 0/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 6/24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 */24 12/24 2/24 
agent 3: 2/24 */24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 21/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 16/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 4/24 6/24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 21/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 16/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 4/24 6/24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 */24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 4/24 6/24 12/24 */24 
agent 4: 0/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 4/24 6/24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 22/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 2/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 2/24 */24 */24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 14/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 14/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 22/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 22/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 */24 22/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 */24 8/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 */24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 8/24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 10/24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 6/24 0/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 10/24 0/24 
agent 3: 2/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 10/24 0/24 
agent 3: 2/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 */24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 3/24 */24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 3/24 */24 
agent 4: 0/24 */24 */24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 7/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 */24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 7/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 */24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 2/24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 2/24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 2/24 9/24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 */24 
agent 3: */24 2/24 9/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 1/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 */24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 9/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 3/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 2/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: */24 4/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 */24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 0/24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 2/24 12/24 4/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 2/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 2/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 3/24 
agent 4: 0/24 */24 */24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 3/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 3/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 3/24 
agent 4: 0/24 */24 12/24 3/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 3/24 
agent 4: 0/24 */24 12/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 3/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 13/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 13/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 3/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 2/24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 2/24 
agent 4: 0/24 */24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 */24 2/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 */24 2/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: 0/24 */24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 3/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 14/24 */24 
agent 4: 0/24 */24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 14/24 */24 
agent 4: 0/24 */24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 4/24 */24 2/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 18/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 */24 2/24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 */24 2/24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 */24 2/24 
agent 4: 0/24 */24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 12/24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 12/24 */24 
agent 4: 0/24 */24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 10/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 2/24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 10/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 2/24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 2/24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 2/24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 */24 
agent 2: */24 */24 2/24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 8/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: */24 8/24 2/24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 8/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 8/24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 15/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 15/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 3/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 9/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 9/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 15/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 15/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 */24 3/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 */24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 8/24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: 0/24 9/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 7/24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 7/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: 0/24 9/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 3/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 */24 1/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 6/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 6/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 20/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 3/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 10/24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 18/24 0/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 18/24 0/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 18/24 0/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 10/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 3/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 3/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 8/24 
agent 4: */24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 8/24 
agent 4: */24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 */24 12/24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: */24 */24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 2/24 */24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 6/24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 4/24 
agent 4: 0/24 12/24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 4/24 
agent 4: 0/24 12/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 4/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 8/24 4/24 
agent 2: 8/24 4/24 8/24 4/24 
agent 3: 8/24 4/24 8/24 4/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 8/24 */24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 */24 8/24 */24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > c > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 */24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 6/24 5/24 5/24 
agent 3: 8/24 6/24 5/24 5/24 
agent 4: 0/24 0/24 14/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 4/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 6/24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 4/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 4/24 */24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 4/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 12/24 6/24 */24 
agent 3: 2/24 12/24 */24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 6/24 5/24 5/24 
agent 3: 8/24 6/24 5/24 5/24 
agent 4: 0/24 0/24 10/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 9/24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 5/24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 9/24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 9/24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 5/24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 5/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 5/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 7/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 17/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 */24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 7/24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 18/24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 4/24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 5/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 18/24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 */24 1/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 */24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 */24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 */24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 5/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 5/24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 5/24 19/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 */24 
agent 4: 0/24 0/24 5/24 19/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 9/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 10/24 0/24 */24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 6/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 3/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 */24 6/24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 21/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 21/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 3/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 */24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 10/24 3/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 21/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 3/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 */24 
agent 2: 12/24 9/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 */24 
agent 2: 12/24 9/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 10/24 0/24 */24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 3/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 0/24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 18/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 3/24 
agent 4: 0/24 */24 */24 3/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 10/24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 3/24 
agent 4: 0/24 */24 */24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 3/24 
agent 4: 0/24 */24 12/24 3/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 3/24 
agent 4: 0/24 */24 */24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 */24 */24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 10/24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 11/24 */24 3/24 
agent 4: 0/24 */24 */24 3/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 */24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 10/24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 10/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 4/24 6/24 
agent 2: 8/24 */24 10/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 10/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 */24 */24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 10/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 18/24 0/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 10/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 */24 */24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 */24 */24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 */24 1/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 */24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 */24 1/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 11/24 1/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 */24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 1/24 */24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 11/24 1/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 4/24 
agent 3: */24 8/24 0/24 */24 
agent 4: 0/24 8/24 */24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 0/24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 0/24 */24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 10/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 0/24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 10/24 
agent 3: */24 8/24 */24 4/24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 6/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 6/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 8/24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 8/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 12/24 8/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 0/24 8/24 */24 
agent 4: 0/24 12/24 8/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 0/24 8/24 */24 
agent 4: 0/24 12/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 0/24 8/24 */24 
agent 4: 0/24 12/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 0/24 8/24 */24 
agent 4: 0/24 12/24 8/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 4/24 
agent 4: 4/24 0/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 0/24 */24 
agent 4: 0/24 8/24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 0/24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 2/24 6/24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 0/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 2/24 6/24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: */24 */24 2/24 6/24 
agent 3: */24 */24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 4/24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 4/24 0/24 */24 
agent 4: 0/24 */24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 4/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 11/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 8/24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 4/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 11/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 4/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 16/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 16/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 16/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 16/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 16/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 16/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 4/24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 */24 6/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 16/24 6/24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 16/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 4/24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 5/24 */24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 0/24 7/24 17/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 6/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 5/24 */24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 0/24 7/24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 5/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 7/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 */24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 */24 
agent 4: 0/24 0/24 */24 19/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 5/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 4/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 */24 
agent 4: 0/24 0/24 */24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 11/24 */24 
agent 4: 0/24 */24 5/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 5/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 */24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 11/24 */24 
agent 4: 0/24 */24 5/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 5/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 1/24 
agent 4: 0/24 0/24 */24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 5/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 11/24 
agent 4: 0/24 */24 6/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 */24 1/24 
agent 4: 0/24 0/24 */24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 */24 1/24 
agent 4: 0/24 0/24 5/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 6/24 18/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 6/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 6/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 6/24 18/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: 0/24 0/24 6/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 6/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 9/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 10/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 10/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: 11/24 10/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 10/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 3/24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 */24 
agent 2: */24 */24 0/24 3/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: 11/24 10/24 3/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 3/24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 1/24 20/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 1/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 3/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 3/24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 3/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 3/24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 3/24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 3/24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 */24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 */24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: */24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 3/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 */24 0/24 3/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 */24 0/24 3/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 */24 0/24 3/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 */24 6/24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 3/24 
agent 3: */24 6/24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 */24 6/24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 */24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 */24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 5/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 11/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 2/24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 11/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 11/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 19/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 19/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 19/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 19/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 19/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 19/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 4/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 19/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 19/24 0/24 5/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 2/24 */24 */24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 8/24 5/24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 19/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: */24 2/24 */24 */24 
agent 3: */24 2/24 9/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 */24 8/24 5/24 
agent 4: 0/24 */24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 11/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 19/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 19/24 0/24 5/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 19/24 0/24 5/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 19/24 0/24 5/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 */24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 19/24 0/24 5/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 19/24 0/24 5/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 19/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 19/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 19/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 19/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 19/24 5/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 19/24 5/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 1/24 */24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: */24 0/24 19/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: */24 4/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: */24 0/24 19/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 19/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 1/24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 19/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 1/24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 19/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 19/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 1/24 0/24 19/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 1/24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 1/24 0/24 19/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 1/24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 1/24 0/24 19/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 1/24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 1/24 0/24 19/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 1/24 4/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 1/24 0/24 19/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 1/24 4/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 */24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 1/24 4/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 19/24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 3/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 3/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 1/24 */24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 3/24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 1/24 */24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 1/24 */24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 4/24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 4/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 1/24 19/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 1/24 19/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 19/24 4/24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 19/24 4/24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 1/24 */24 4/24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 19/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 1/24 */24 4/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 1/24 11/24 4/24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 11/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 0/24 */24 */24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 0/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 */24 
agent 3: */24 0/24 11/24 */24 
agent 4: 0/24 */24 5/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 */24 
agent 3: */24 0/24 11/24 */24 
agent 4: 0/24 */24 5/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 4/24 */24 
agent 3: */24 0/24 11/24 */24 
agent 4: 0/24 */24 5/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 11/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 0/24 11/24 5/24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 19/24 5/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 19/24 5/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 19/24 5/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 2/24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 1/24 4/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 4/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 4/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 0/24 4/24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 5/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 */24 4/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 19/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 5/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 */24 4/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 */24 4/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 1/24 4/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 19/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 0/24 1/24 4/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 19/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 1/24 4/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 1/24 0/24 19/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 10/24 
agent 3: 1/24 8/24 */24 4/24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 10/24 
agent 3: 1/24 8/24 */24 4/24 
agent 4: 0/24 8/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 6/24 8/24 0/24 10/24 
agent 3: 1/24 8/24 11/24 4/24 
agent 4: 0/24 8/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 */24 */24 
agent 2: 6/24 8/24 0/24 10/24 
agent 3: 1/24 8/24 11/24 4/24 
agent 4: 0/24 8/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 11/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: 0/24 8/24 11/24 */24 
agent 4: 0/24 8/24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 0/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: 6/24 8/24 10/24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 7/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 0/24 4/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 10/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 6/24 8/24 */24 4/24 
agent 3: */24 8/24 */24 10/24 
agent 4: 0/24 8/24 6/24 10/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 3/24 */24 
agent 4: 0/24 8/24 2/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 0/24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 6/24 10/24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: */24 */24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 */24 0/24 
agent 3: 0/24 1/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 6/24 8/24 10/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 2/24 8/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: */24 */24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: */24 */24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 6/24 10/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 1/24 11/24 4/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 */24 4/24 8/24 
agent 3: */24 */24 10/24 8/24 
agent 4: 0/24 6/24 10/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 1/24 11/24 4/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 */24 */24 0/24 
agent 2: 6/24 0/24 */24 8/24 
agent 3: 1/24 11/24 4/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 */24 */24 0/24 
agent 2: 6/24 0/24 10/24 8/24 
agent 3: 1/24 11/24 4/24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 10/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 10/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 0/24 2/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 2/24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 2/24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: */24 */24 10/24 8/24 
agent 4: 0/24 6/24 10/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 0/24 
agent 3: 0/24 3/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 0/24 
agent 3: 0/24 3/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 3/24 9/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 6/24 0/24 
agent 3: 0/24 3/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 9/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 */24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 */24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 4/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 16/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: 0/24 3/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 6/24 0/24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 4/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 4/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 12/24 4/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: */24 */24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: 0/24 3/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 3/24 12/24 9/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 9/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 9/24 
agent 4: 0/24 3/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 3/24 12/24 9/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 9/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 */24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 4/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 4/24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 20/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 20/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 4/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: */24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 20/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 20/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 10/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 20/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: */24 10/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: 10/24 10/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 10/24 2/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 10/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 10/24 2/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 10/24 2/24 12/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 10/24 2/24 12/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: */24 */24 7/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 2/24 12/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: */24 3/24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 */24 7/24 */24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 2/24 12/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: */24 3/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 2/24 12/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 2/24 3/24 7/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 10/24 2/24 */24 */24 
agent 3: 2/24 */24 */24 11/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 */24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 2/24 */24 3/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 */24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: 8/24 */24 3/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 */24 0/24 
agent 3: 8/24 */24 3/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 3/24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 1/24 0/24 3/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 */24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 18/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 18/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 7/24 3/24 12/24 
agent 4: 2/24 */24 3/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 2/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 1/24 0/24 3/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 12/24 */24 4/24 
agent 4: 1/24 0/24 3/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: */24 4/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 8/24 */24 3/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 8/24 */24 3/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 18/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: */24 4/24 12/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 18/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 18/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 7/24 12/24 3/24 
agent 4: 2/24 */24 12/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 8/24 3/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 3/24 
agent 2: */24 */24 8/24 3/24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 3/24 
agent 2: */24 */24 8/24 3/24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 3/24 
agent 2: */24 */24 2/24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 7/24 12/24 3/24 
agent 4: 2/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 2/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 */24 12/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 */24 12/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 18/24 2/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 2/24 */24 12/24 3/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 8/24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 8/24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 8/24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 2/24 18/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 2/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 3/24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 3/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 */24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 18/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 3/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: */24 3/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 1/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 3/24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 3/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 1/24 20/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 10/24 3/24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: */24 10/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 10/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 */24 
agent 2: */24 10/24 0/24 */24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: 11/24 10/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 */24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 11/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 */24 
agent 2: */24 2/24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 11/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 17/24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 2/24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 17/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 17/24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 2/24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 11/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 17/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 17/24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 17/24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 7/24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 7/24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 4/24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: */24 0/24 */24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 7/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 9/24 7/24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: */24 9/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: */24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: */24 0/24 */24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 7/24 
agent 4: 0/24 */24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 9/24 8/24 7/24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 9/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 17/24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 17/24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: */24 */24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: */24 */24 12/24 3/24 
agent 4: */24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: */24 */24 12/24 3/24 
agent 4: */24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: */24 */24 12/24 3/24 
agent 4: */24 */24 12/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: 10/24 */24 0/24 4/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: 10/24 */24 0/24 4/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: 10/24 10/24 0/24 4/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: 10/24 10/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 10/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 0/24 6/24 
agent 3: 8/24 4/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 10/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 10/24 0/24 4/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 10/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 4/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 11/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 10/24 */24 */24 4/24 
agent 3: */24 6/24 */24 5/24 
agent 4: */24 0/24 */24 11/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 10/24 */24 */24 4/24 
agent 3: */24 6/24 */24 5/24 
agent 4: */24 0/24 */24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 10/24 */24 */24 4/24 
agent 3: */24 6/24 */24 5/24 
agent 4: */24 0/24 */24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 4/24 
agent 2: */24 16/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 4/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 4/24 
agent 4: */24 */24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 4/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 4/24 0/24 8/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 11/24 
agent 4: 0/24 */24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 4/24 
agent 2: */24 16/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 16/24 6/24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: */24 8/24 8/24 */24 
agent 4: 0/24 8/24 8/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 2/24 0/24 16/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 4/24 
agent 2: 8/24 */24 5/24 4/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: 2/24 8/24 8/24 6/24 
agent 4: 0/24 8/24 8/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 4/24 
agent 2: 8/24 */24 5/24 4/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 7/24 5/24 4/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 7/24 5/24 4/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 5/24 14/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 4/24 8/24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: */24 0/24 */24 12/24 
agent 3: */24 7/24 */24 4/24 
agent 4: */24 */24 */24 4/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 5/24 8/24 5/24 
agent 4: 6/24 0/24 8/24 10/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > c > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 5/24 8/24 5/24 
agent 4: 6/24 0/24 8/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 6/24 5/24 8/24 5/24 
agent 4: 6/24 0/24 8/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 6/24 */24 8/24 */24 
agent 4: */24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 14/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 5/24 14/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 7/24 5/24 4/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 5/24 14/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 7/24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 5/24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 5/24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 5/24 14/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 7/24 5/24 4/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 5/24 14/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 7/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 2/24 0/24 16/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 2/24 0/24 16/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 */24 6/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 12/24 4/24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 2/24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: */24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: */24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 2/24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 2/24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 2/24 0/24 16/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 */24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 */24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 2/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 16/24 6/24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: */24 0/24 11/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 8/24 6/24 8/24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: */24 0/24 11/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 4/24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: */24 0/24 11/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 10/24 4/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 10/24 4/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 10/24 4/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 2/24 0/24 10/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 10/24 */24 4/24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: */24 0/24 11/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: 2/24 */24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 6/24 0/24 
agent 2: 12/24 */24 6/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 */24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 */24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 6/24 0/24 
agent 2: 12/24 */24 6/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 6/24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 6/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 6/24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 4/24 6/24 8/24 
agent 3: */24 9/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 4/24 0/24 
agent 2: 10/24 10/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 10/24 4/24 0/24 
agent 2: 10/24 10/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 10/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 2/24 0/24 
agent 2: 10/24 2/24 12/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: */24 */24 7/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 */24 
agent 2: 10/24 10/24 0/24 4/24 
agent 3: */24 4/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 2/24 0/24 
agent 2: 10/24 2/24 12/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: */24 */24 7/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 2/24 12/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: */24 */24 7/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 2/24 12/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 2/24 3/24 7/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: */24 */24 12/24 3/24 
agent 4: */24 */24 12/24 7/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 */24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 2/24 */24 3/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 */24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 2/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 1/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 */24 0/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 1/24 0/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 */24 12/24 */24 
agent 4: 0/24 1/24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 11/24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 11/24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 2/24 11/24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: 11/24 2/24 11/24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 3/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: */24 */24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 10/24 0/24 3/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: */24 */24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: */24 */24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 */24 0/24 
agent 2: 11/24 2/24 11/24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 */24 0/24 
agent 2: 11/24 2/24 11/24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 11/24 0/24 
agent 3: */24 3/24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 11/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 */24 0/24 
agent 2: 11/24 10/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 */24 0/24 
agent 2: 11/24 10/24 */24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 10/24 3/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 10/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 3/24 0/24 
agent 2: 1/24 20/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 */24 
agent 2: */24 10/24 0/24 */24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 11/24 */24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 9/24 */24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 3/24 0/24 
agent 2: 1/24 20/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 */24 0/24 0/24 
agent 2: 1/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 3/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 */24 0/24 0/24 
agent 2: 1/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 3/24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: 2/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 3/24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 3/24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 2/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 2/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: */24 20/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: */24 20/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 10/24 0/24 */24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 3/24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 10/24 0/24 */24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 10/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 10/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 */24 
agent 2: 11/24 10/24 0/24 */24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 2/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 */24 0/24 3/24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 */24 0/24 3/24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 3/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 9/24 0/24 */24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 0/24 */24 
agent 2: */24 */24 0/24 3/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 9/24 */24 
agent 3: 0/24 */24 */24 5/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 7/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 15/24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 18/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 */24 
agent 4: 0/24 18/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 14/24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 7/24 3/24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 14/24 */24 
agent 3: 8/24 0/24 */24 15/24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 7/24 3/24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 11/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 */24 */24 2/24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 9/24 */24 
agent 3: 0/24 */24 */24 5/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 11/24 11/24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 9/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 6/24 0/24 */24 1/24 
agent 4: */24 0/24 1/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 1/24 
agent 4: 0/24 8/24 2/24 14/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 3/24 */24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 7/24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 23/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 23/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 15/24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 15/24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 1/24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 3/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 */24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 3/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 */24 8/24 1/24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 1/24 
agent 4: 0/24 8/24 2/24 14/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 1/24 
agent 4: */24 0/24 1/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 7/24 3/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 7/24 3/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 1/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 */24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 1/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 0/24 1/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 */24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 3/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 0/24 1/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 3/24 
agent 3: 0/24 6/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 1/24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 0/24 1/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 1/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 1/24 */24 
agent 4: 0/24 */24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 12/24 1/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 */24 1/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 11/24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 1/24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 1/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 3/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 9/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 9/24 12/24 */24 
agent 4: 0/24 */24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 9/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 1/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 1/24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 1/24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 15/24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 0/24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 0/24 1/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 1/24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 1/24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 0/24 1/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 1/24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 3/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 0/24 1/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 1/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: 0/24 */24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 */24 0/24 
agent 2: 11/24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 12/24 */24 0/24 
agent 2: 11/24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 12/24 1/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 12/24 1/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 1/24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 */24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 */24 0/24 
agent 2: 11/24 10/24 3/24 0/24 
agent 3: 2/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 1/24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 3/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 3/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 */24 
agent 2: 12/24 */24 0/24 3/24 
agent 3: 0/24 6/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 */24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 */24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 0/24 3/24 
agent 3: 0/24 6/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 12/24 0/24 1/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 0/24 3/24 
agent 3: 0/24 6/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 12/24 0/24 1/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 1/24 
agent 2: */24 0/24 */24 1/24 
agent 3: */24 */24 1/24 */24 
agent 4: 0/24 */24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 1/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 7/24 3/24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 13/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 7/24 3/24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 */24 12/24 
agent 4: 0/24 11/24 1/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 8/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 */24 9/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 3/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 */24 9/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 3/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 1/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 3/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 1/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 0/24 */24 
agent 2: 8/24 */24 */24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 15/24 0/24 */24 
agent 2: 8/24 */24 */24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 0/24 1/24 
agent 2: 8/24 */24 */24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 0/24 1/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: */24 */24 12/24 3/24 
agent 4: */24 3/24 12/24 7/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 4/24 12/24 2/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 1/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 13/24 
agent 4: 0/24 */24 18/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 1/24 8/24 12/24 3/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 1/24 12/24 8/24 3/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 1/24 0/24 
agent 2: 11/24 */24 1/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 1/24 0/24 
agent 2: 11/24 */24 1/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 3/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 3/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 3/24 8/24 */24 
agent 3: 0/24 */24 8/24 1/24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 1/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 */24 
agent 2: 1/24 8/24 12/24 3/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 3/24 */24 
agent 3: 0/24 8/24 */24 1/24 
agent 4: 0/24 8/24 2/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 1/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 */24 
agent 2: 1/24 8/24 12/24 3/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 1/24 8/24 12/24 3/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: */24 4/24 12/24 2/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 1/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 1/24 12/24 8/24 3/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 3/24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: */24 4/24 12/24 2/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 1/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 1/24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 1/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 2/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 */24 12/24 4/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 1/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 2/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 3/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 13/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 */24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 2/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 2/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 21/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 21/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 13/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 13/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 8/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 3/24 8/24 
agent 4: 0/24 0/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 3/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 */24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 3/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 3/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 18/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 18/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 3/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 2/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 7/24 12/24 3/24 
agent 4: */24 */24 12/24 3/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 4/24 12/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 */24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 18/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 3/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 3/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 */24 
agent 3: 1/24 8/24 12/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 */24 */24 3/24 
agent 3: */24 6/24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 1/24 12/24 8/24 3/24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 */24 */24 3/24 
agent 3: */24 6/24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 */24 */24 3/24 
agent 3: */24 6/24 */24 5/24 
agent 4: 0/24 0/24 */24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 */24 */24 3/24 
agent 3: */24 6/24 */24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 3/24 
agent 2: */24 */24 */24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 3/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 11/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 8/24 */24 3/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 0/24 11/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 3/24 
agent 2: */24 */24 */24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 3/24 
agent 2: */24 */24 8/24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 */24 0/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 3/24 
agent 2: */24 */24 8/24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 8/24 3/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 8/24 3/24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 6/24 6/24 
agent 4: 0/24 10/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 3/24 
agent 2: */24 */24 8/24 3/24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 10/24 0/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 8/24 3/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 0/24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 8/24 8/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 */24 
agent 4: */24 0/24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 8/24 */24 
agent 2: */24 8/24 8/24 3/24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 1/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 8/24 */24 
agent 2: */24 8/24 8/24 3/24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 8/24 */24 
agent 2: 5/24 8/24 8/24 3/24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 8/24 */24 
agent 2: 5/24 8/24 8/24 3/24 
agent 3: */24 8/24 8/24 */24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 3/24 
agent 3: 0/24 8/24 */24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 5/24 8/24 3/24 */24 
agent 4: 0/24 8/24 2/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 3/24 
agent 3: 0/24 8/24 */24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 2/24 
agent 2: */24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 2/24 
agent 2: 6/24 12/24 */24 2/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 8/24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 8/24 */24 
agent 2: 5/24 8/24 8/24 3/24 
agent 3: */24 8/24 8/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 13/24 */24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 */24 2/24 
agent 2: 6/24 12/24 */24 2/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 */24 
agent 3: */24 0/24 */24 1/24 
agent 4: */24 0/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 2/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 2/24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 2/24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 2/24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 2/24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 2/24 
agent 2: */24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 3/24 
agent 4: 0/24 0/24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 3/24 
agent 3: 1/24 8/24 12/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 3/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 */24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 3/24 
agent 4: 0/24 0/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 8/24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: */24 */24 8/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: */24 */24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 */24 8/24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: */24 */24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: */24 */24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: */24 */24 8/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 1/24 3/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 3/24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: 1/24 3/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 3/24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 */24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 8/24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 8/24 3/24 
agent 3: 0/24 */24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 3/24 
agent 3: */24 */24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 13/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 5/24 
agent 4: 0/24 0/24 */24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 11/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 11/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 0/24 9/24 */24 
agent 3: 0/24 */24 */24 5/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 3/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 6/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 */24 
agent 2: 11/24 10/24 0/24 3/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: */24 */24 2/24 6/24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 */24 
agent 3: 0/24 */24 */24 5/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 */24 
agent 3: 0/24 */24 */24 5/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 */24 */24 5/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 */24 8/24 5/24 
agent 4: 0/24 */24 0/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 8/24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 3/24 
agent 4: 0/24 */24 */24 3/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 18/24 1/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 11/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 8/24 */24 3/24 
agent 3: */24 */24 */24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 11/24 11/24 
agent 4: 0/24 */24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 13/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 3/24 8/24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 3/24 8/24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: 0/24 0/24 13/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 3/24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 11/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: */24 */24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 3/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 3/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: 1/24 3/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 6/24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 */24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 */24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 2/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 3/24 8/24 */24 
agent 3: 0/24 */24 8/24 1/24 
agent 4: 0/24 */24 8/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 3/24 8/24 */24 
agent 3: 0/24 */24 8/24 1/24 
agent 4: 0/24 */24 8/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 3/24 8/24 */24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 4/24 0/24 */24 
agent 2: */24 3/24 8/24 */24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 4/24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 1/24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 2/24 8/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 4/24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 4/24 */24 0/24 
agent 2: */24 3/24 */24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 1/24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 2/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 1/24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: */24 */24 */24 1/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 3/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 5/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 5/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 5/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 4/24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 2/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 */24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: d > c > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 */24 6/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 */24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 3/24 */24 
agent 3: */24 */24 3/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 13/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 13/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 */24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 3/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 */24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 3/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 14/24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 14/24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 5/24 */24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 5/24 */24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 10/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: 2/24 11/24 9/24 2/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 6/24 0/24 */24 1/24 
agent 4: */24 0/24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 6/24 0/24 */24 1/24 
agent 4: */24 0/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 6/24 0/24 */24 1/24 
agent 4: */24 0/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: */24 0/24 1/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 */24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: */24 0/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 14/24 */24 
agent 3: 8/24 0/24 */24 15/24 
agent 4: 0/24 */24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 9/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 9/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 11/24 11/24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 0/24 9/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 11/24 11/24 */24 
agent 4: 0/24 11/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 */24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 7/24 3/24 
agent 2: 8/24 0/24 14/24 */24 
agent 3: 8/24 0/24 */24 15/24 
agent 4: 0/24 */24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 */24 9/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 7/24 3/24 
agent 2: 8/24 0/24 14/24 */24 
agent 3: 8/24 0/24 */24 15/24 
agent 4: 0/24 */24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 2/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 13/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 13/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 9/24 */24 
agent 3: 0/24 11/24 8/24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 11/24 */24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 8/24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 9/24 */24 
agent 3: 0/24 11/24 8/24 */24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 */24 0/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 7/24 3/24 
agent 2: 8/24 0/24 13/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 0/24 */24 9/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 11/24 */24 5/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 7/24 3/24 
agent 2: 8/24 0/24 13/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 3/24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 */24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 21/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 1/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 1/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 3/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: 1/24 20/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: 1/24 20/24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 10/24 3/24 0/24 
agent 2: 11/24 10/24 3/24 0/24 
agent 3: 2/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 10/24 3/24 0/24 
agent 2: 11/24 10/24 3/24 0/24 
agent 3: 2/24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 10/24 3/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 3/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 */24 
agent 2: */24 */24 0/24 3/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 11/24 3/24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 3/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 3/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: */24 2/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 */24 0/24 
agent 3: 2/24 3/24 */24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 11/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 11/24 */24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 */24 0/24 
agent 2: 11/24 2/24 11/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 2/24 0/24 */24 
agent 3: */24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 2/24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 */24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 */24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 18/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 18/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 */24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 18/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 3/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 */24 18/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 2/24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 2/24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 5/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 5/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 11/24 */24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 11/24 */24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 */24 14/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 0/24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 4/24 14/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 0/24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 5/24 5/24 6/24 
agent 4: 0/24 4/24 14/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: */24 */24 6/24 5/24 
agent 3: 8/24 5/24 6/24 5/24 
agent 4: 0/24 4/24 6/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 5/24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 0/24 6/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 4/24 14/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: */24 */24 6/24 5/24 
agent 3: 8/24 5/24 6/24 5/24 
agent 4: 0/24 4/24 6/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 5/24 6/24 5/24 
agent 4: 0/24 4/24 6/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 0/24 6/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 4/24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 0/24 6/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 4/24 14/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 0/24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 5/24 5/24 6/24 
agent 4: 0/24 4/24 14/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 5/24 */24 6/24 
agent 4: 0/24 */24 11/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 5/24 */24 6/24 
agent 4: 0/24 */24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 */24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 2/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 19/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 1/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 7/24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 7/24 */24 
agent 4: 0/24 11/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 7/24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 */24 7/24 
agent 4: 0/24 */24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 */24 */24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 7/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 9/24 6/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 */24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 17/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 17/24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 */24 11/24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 */24 
agent 4: 0/24 0/24 17/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 0/24 */24 8/24 
agent 4: 0/24 */24 4/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 0/24 */24 8/24 
agent 4: 0/24 */24 4/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 0/24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 0/24 */24 8/24 
agent 4: 0/24 12/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 0/24 */24 8/24 
agent 4: 0/24 12/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 0/24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: 2/24 8/24 6/24 8/24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: 2/24 8/24 6/24 8/24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 8/24 6/24 8/24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 8/24 6/24 8/24 
agent 3: 2/24 8/24 6/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 */24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: */24 8/24 8/24 */24 
agent 4: 0/24 8/24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 8/24 8/24 
agent 3: 2/24 6/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: */24 */24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 8/24 8/24 
agent 3: 2/24 6/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 8/24 */24 8/24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 6/24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 6/24 8/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: */24 8/24 */24 8/24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 8/24 */24 8/24 
agent 3: */24 8/24 */24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 8/24 6/24 8/24 
agent 3: 2/24 8/24 6/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 6/24 0/24 10/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: 2/24 8/24 8/24 6/24 
agent 4: 0/24 8/24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 6/24 8/24 6/24 4/24 
agent 3: */24 8/24 */24 10/24 
agent 4: 0/24 8/24 6/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 6/24 0/24 10/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 5/24 8/24 
agent 3: */24 */24 5/24 8/24 
agent 4: 6/24 0/24 10/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 5/24 14/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 5/24 */24 
agent 3: 6/24 */24 */24 */24 
agent 4: */24 0/24 7/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 5/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 5/24 5/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 7/24 4/24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 5/24 5/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 12/24 */24 
agent 3: */24 7/24 4/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 5/24 8/24 
agent 3: */24 5/24 5/24 8/24 
agent 4: 6/24 0/24 10/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 5/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 4/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 */24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 4/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 4/24 */24 
agent 4: 0/24 */24 4/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 4/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 */24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 */24 17/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 */24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 */24 17/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 6/24 1/24 
agent 4: 0/24 */24 4/24 19/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 2/24 */24 6/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 6/24 1/24 
agent 4: 0/24 */24 4/24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 */24 
agent 4: 0/24 */24 */24 19/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 1/24 4/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 4/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 */24 
agent 4: 0/24 */24 */24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 */24 */24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 17/24 */24 1/24 
agent 4: 0/24 */24 */24 19/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 17/24 */24 1/24 
agent 4: 0/24 */24 */24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 */24 */24 19/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 6/24 11/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 6/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 7/24 6/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 2/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 11/24 
agent 4: 0/24 */24 6/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: */24 6/24 6/24 */24 
agent 3: */24 5/24 6/24 */24 
agent 4: */24 5/24 6/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 11/24 
agent 4: 0/24 */24 6/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 */24 6/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 7/24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 7/24 */24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 7/24 6/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 7/24 */24 11/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 */24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 2/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 */24 6/24 5/24 
agent 4: 0/24 4/24 6/24 14/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 */24 4/24 */24 
agent 3: */24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 */24 6/24 5/24 
agent 4: 0/24 4/24 6/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 8/24 5/24 6/24 5/24 
agent 4: 0/24 4/24 6/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 5/24 6/24 5/24 
agent 4: 0/24 4/24 6/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 2/24 0/24 */24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: */24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 2/24 0/24 */24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: */24 12/24 */24 */24 
agent 3: */24 12/24 */24 4/24 
agent 4: 2/24 0/24 6/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 18/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 */24 */24 9/24 
agent 4: 0/24 */24 */24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 */24 */24 9/24 
agent 4: 0/24 */24 */24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 10/24 4/24 10/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 10/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 5/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 10/24 4/24 10/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 4/24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 */24 */24 9/24 
agent 4: 0/24 11/24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 10/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 5/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 11/24 
agent 4: 0/24 */24 6/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 11/24 
agent 4: 0/24 */24 6/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 11/24 
agent 4: 0/24 */24 6/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 3/24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 1/24 4/24 */24 
agent 4: 0/24 */24 6/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 0/24 
agent 2: 8/24 3/24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 1/24 12/24 
agent 3: */24 */24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 1/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 2/24 22/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 */24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 12/24 1/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: */24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: */24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: */24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: */24 */24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 4/24 12/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: */24 12/24 */24 6/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 */24 12/24 */24 
agent 4: 4/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 2/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 6/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 6/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 6/24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 6/24 12/24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 */24 */24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 6/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 17/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 6/24 6/24 
agent 4: 0/24 10/24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 */24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 20/24 4/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: */24 2/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 */24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 1/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 1/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 */24 */24 
agent 4: 0/24 1/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 */24 */24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 */24 
agent 3: 0/24 19/24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 4/24 0/24 */24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 1/24 0/24 19/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 */24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 */24 
agent 3: 0/24 19/24 0/24 */24 
agent 4: 0/24 1/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 */24 0/24 5/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 1/24 */24 5/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 */24 0/24 5/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 11/24 */24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 2/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 11/24 */24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 1/24 */24 6/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 17/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 */24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 */24 7/24 
agent 4: 0/24 */24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 */24 */24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 */24 
agent 4: 0/24 0/24 */24 7/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 9/24 6/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 */24 
agent 4: 0/24 0/24 */24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 17/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 17/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 4/24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 */24 4/24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 1/24 4/24 19/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 17/24 6/24 */24 
agent 4: 0/24 */24 4/24 19/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 */24 1/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 1/24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 1/24 */24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: */24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: */24 19/24 0/24 */24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 */24 
agent 3: 0/24 19/24 */24 */24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 */24 
agent 3: 0/24 19/24 */24 */24 
agent 4: 0/24 */24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 */24 */24 
agent 4: 0/24 */24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 */24 4/24 
agent 4: 0/24 1/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: */24 19/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 */24 4/24 
agent 4: 0/24 1/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 17/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: */24 19/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 19/24 0/24 4/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 */24 0/24 */24 
agent 2: 6/24 0/24 8/24 */24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: 0/24 */24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 */24 0/24 */24 
agent 2: 6/24 0/24 8/24 10/24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 */24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 */24 
agent 3: */24 9/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 4/24 
agent 3: */24 0/24 8/24 */24 
agent 4: 0/24 12/24 8/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 */24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 2/24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 */24 2/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 */24 2/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 11/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 */24 11/24 */24 
agent 4: 0/24 */24 11/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 6/24 4/24 8/24 6/24 
agent 3: */24 9/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 5/24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 11/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 11/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 5/24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 5/24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 2/24 */24 
agent 3: 2/24 5/24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 7/24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 */24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 5/24 */24 6/24 
agent 4: 0/24 */24 11/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 */24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 7/24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 7/24 11/24 6/24 
agent 4: 0/24 7/24 11/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 7/24 11/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 7/24 11/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 7/24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 10/24 4/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 6/24 8/24 4/24 6/24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 7/24 */24 
agent 4: 0/24 11/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 7/24 11/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 7/24 11/24 6/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 18/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 7/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 17/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 17/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 */24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 9/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 9/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 9/24 8/24 7/24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 6/24 8/24 0/24 */24 
agent 3: 0/24 8/24 11/24 */24 
agent 4: 0/24 8/24 */24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 6/24 8/24 4/24 6/24 
agent 3: */24 8/24 9/24 */24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 9/24 8/24 7/24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 0/24 9/24 8/24 7/24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 9/24 7/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 6/24 0/24 8/24 */24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 9/24 7/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: 0/24 9/24 7/24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 6/24 10/24 8/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 0/24 */24 8/24 
agent 3: 0/24 11/24 5/24 8/24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 6/24 10/24 8/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 0/24 6/24 10/24 8/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 */24 */24 */24 
agent 3: 0/24 6/24 10/24 8/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 */24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 */24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 */24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 10/24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 10/24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: 6/24 8/24 */24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 */24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 0/24 
agent 2: 6/24 8/24 */24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 6/24 8/24 10/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 6/24 0/24 
agent 2: 6/24 12/24 6/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 6/24 8/24 0/24 */24 
agent 3: 0/24 8/24 11/24 */24 
agent 4: 0/24 8/24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 2/24 10/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 6/24 0/24 
agent 2: 6/24 12/24 6/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 6/24 0/24 
agent 2: 6/24 12/24 6/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 6/24 0/24 
agent 2: 6/24 12/24 6/24 0/24 
agent 3: 6/24 0/24 6/24 12/24 
agent 4: 6/24 0/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 0/24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 6/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 6/24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 7/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 4/24 12/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 1/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: */24 4/24 12/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 6/24 4/24 12/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: 6/24 */24 12/24 */24 
agent 3: 6/24 4/24 12/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 8/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 */24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 2/24 
agent 2: 6/24 12/24 */24 */24 
agent 3: */24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: 6/24 */24 12/24 */24 
agent 4: */24 1/24 0/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 2/24 
agent 2: 6/24 12/24 */24 2/24 
agent 3: */24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 8/24 8/24 */24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 */24 
agent 4: */24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 */24 
agent 4: */24 0/24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 8/24 2/24 
agent 2: 5/24 8/24 8/24 3/24 
agent 3: */24 8/24 8/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 2/24 
agent 2: 6/24 */24 12/24 */24 
agent 3: 6/24 4/24 12/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 8/24 2/24 
agent 2: 5/24 8/24 8/24 3/24 
agent 3: */24 8/24 8/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 8/24 2/24 
agent 2: 5/24 8/24 8/24 3/24 
agent 3: 5/24 8/24 8/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 2/24 
agent 2: */24 */24 */24 3/24 
agent 3: */24 */24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 0/24 2/24 
agent 2: */24 */24 */24 3/24 
agent 3: */24 */24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 8/24 0/24 2/24 
agent 2: */24 */24 */24 3/24 
agent 3: */24 */24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 8/24 0/24 2/24 
agent 2: 5/24 8/24 8/24 3/24 
agent 3: 5/24 8/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 2/24 
agent 2: 1/24 12/24 8/24 3/24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 8/24 */24 0/24 
agent 2: */24 */24 3/24 */24 
agent 3: */24 */24 3/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 8/24 */24 3/24 
agent 3: */24 6/24 */24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 1/24 
agent 3: 5/24 3/24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 2/24 
agent 2: 1/24 12/24 8/24 3/24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 1/24 12/24 8/24 3/24 
agent 3: */24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 1/24 12/24 8/24 3/24 
agent 3: 1/24 12/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 8/24 12/24 3/24 
agent 3: 1/24 8/24 12/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 8/24 12/24 3/24 
agent 3: 1/24 8/24 12/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 1/24 8/24 12/24 3/24 
agent 3: 1/24 8/24 12/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 8/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 8/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 */24 12/24 
agent 4: */24 8/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 1/24 8/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 2/24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 */24 8/24 12/24 
agent 4: 1/24 3/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 3/24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 2/24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 16/24 2/24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 16/24 2/24 0/24 
agent 2: 6/24 0/24 18/24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 4/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 16/24 0/24 2/24 
agent 2: 6/24 */24 12/24 */24 
agent 3: 6/24 4/24 12/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 16/24 0/24 2/24 
agent 2: 6/24 */24 12/24 */24 
agent 3: 6/24 4/24 12/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 16/24 0/24 2/24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: 6/24 4/24 12/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 13/24 */24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 4/24 2/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 3/24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 4/24 2/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 3/24 0/24 
agent 2: 8/24 0/24 16/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 3/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 7/24 3/24 12/24 
agent 4: 2/24 */24 3/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: */24 7/24 3/24 12/24 
agent 4: 2/24 */24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 2/24 7/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 4/24 0/24 
agent 2: */24 0/24 14/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 2/24 7/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 0/24 14/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 2/24 7/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 7/24 12/24 3/24 
agent 4: */24 */24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 10/24 */24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 10/24 0/24 14/24 0/24 
agent 3: */24 7/24 3/24 12/24 
agent 4: 2/24 */24 3/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 */24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: 0/24 */24 12/24 9/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 */24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 2/24 9/24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 2/24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 2/24 9/24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 2/24 9/24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 2/24 0/24 16/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 2/24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 16/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 16/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 7/24 11/24 6/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 0/24 2/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 2/24 0/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 2/24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 16/24 0/24 6/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 0/24 2/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: */24 0/24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 0/24 
agent 2: 10/24 0/24 14/24 0/24 
agent 3: */24 7/24 3/24 12/24 
agent 4: 2/24 */24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 */24 0/24 
agent 2: 10/24 0/24 14/24 0/24 
agent 3: */24 7/24 3/24 12/24 
agent 4: 2/24 */24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 */24 */24 0/24 
agent 2: 10/24 0/24 14/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 2/24 7/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 0/24 14/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 2/24 7/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 2/24 */24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 10/24 0/24 12/24 */24 
agent 3: */24 7/24 4/24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: */24 4/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 */24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: */24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: */24 4/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 2/24 */24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 12/24 */24 */24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 2/24 */24 12/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 2/24 7/24 12/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 2/24 7/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 8/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 10/24 0/24 */24 12/24 
agent 3: */24 7/24 */24 4/24 
agent 4: */24 */24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: */24 7/24 12/24 3/24 
agent 4: */24 */24 12/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 8/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 11/24 8/24 */24 
agent 4: */24 11/24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 8/24 4/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 */24 8/24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 3/24 
agent 4: 0/24 11/24 */24 3/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: 2/24 11/24 9/24 2/24 
agent 4: */24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 8/24 4/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 */24 8/24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 8/24 4/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 */24 8/24 */24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 8/24 4/24 
agent 2: 8/24 4/24 8/24 4/24 
agent 3: 8/24 4/24 8/24 4/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 3/24 
agent 2: */24 */24 8/24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 4/24 */24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 */24 5/24 */24 
agent 4: 0/24 0/24 14/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 0/24 4/24 
agent 2: 8/24 6/24 5/24 5/24 
agent 3: 8/24 6/24 5/24 5/24 
agent 4: 0/24 0/24 14/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 8/24 */24 3/24 
agent 3: */24 6/24 */24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 10/24 */24 5/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 0/24 10/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 6/24 5/24 5/24 
agent 3: 8/24 6/24 5/24 5/24 
agent 4: 0/24 0/24 10/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 5/24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 7/24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 6/24 */24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 5/24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 7/24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 5/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 7/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: */24 8/24 */24 3/24 
agent 3: */24 6/24 */24 5/24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 0/24 3/24 
agent 2: */24 8/24 */24 3/24 
agent 3: */24 6/24 */24 5/24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 8/24 */24 3/24 
agent 3: */24 6/24 */24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 8/24 3/24 
agent 2: */24 */24 8/24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 10/24 */24 */24 4/24 
agent 3: */24 6/24 */24 5/24 
agent 4: */24 0/24 11/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 8/24 3/24 
agent 2: */24 */24 8/24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: */24 */24 8/24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: */24 */24 2/24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 4/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 4/24 
agent 4: */24 10/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 0/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 13/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 10/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 10/24 0/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 13/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 7/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 10/24 0/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: */24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 7/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 10/24 0/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: */24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 12/24 12/24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 */24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 17/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 17/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 */24 
agent 4: 0/24 0/24 17/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 10/24 1/24 5/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 17/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 */24 5/24 
agent 4: 0/24 0/24 17/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 1/24 5/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 1/24 */24 6/24 
agent 3: 8/24 0/24 */24 7/24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 1/24 5/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 10/24 1/24 5/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 */24 
agent 2: 10/24 10/24 0/24 4/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: */24 0/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 */24 
agent 2: 10/24 10/24 0/24 4/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: */24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 10/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 10/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 2/24 0/24 12/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 4/24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 */24 6/24 */24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: */24 12/24 0/24 */24 
agent 3: 2/24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 */24 10/24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 2/24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: */24 12/24 0/24 */24 
agent 3: 2/24 0/24 12/24 */24 
agent 4: */24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: */24 12/24 0/24 */24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 0/24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 7/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 17/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 */24 10/24 
agent 4: 0/24 2/24 17/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 */24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 3/24 
agent 2: */24 */24 2/24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: 6/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: 6/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 */24 
agent 4: 6/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: 6/24 0/24 1/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 0/24 */24 4/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 9/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: 6/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: 6/24 0/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: 6/24 0/24 1/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: 6/24 0/24 1/24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 0/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 10/24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 5/24 6/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 5/24 6/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 4/24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 */24 6/24 5/24 
agent 4: */24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 */24 4/24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 */24 6/24 5/24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 */24 */24 4/24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 */24 6/24 5/24 
agent 4: */24 */24 0/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 */24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 7/24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 7/24 4/24 
agent 2: */24 */24 */24 4/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 2/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 */24 6/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 8/24 5/24 5/24 
agent 3: */24 */24 0/24 */24 
agent 4: */24 0/24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 14/24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 */24 2/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 9/24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 */24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 9/24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 */24 2/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 9/24 2/24 
agent 2: */24 */24 3/24 */24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 */24 2/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 3/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 3/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 */24 2/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 9/24 2/24 
agent 2: */24 0/24 3/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 14/24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 14/24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 7/24 */24 
agent 2: 8/24 5/24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 7/24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 7/24 4/24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 0/24 11/24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 */24 10/24 
agent 4: 0/24 */24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 */24 */24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 4/24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 1/24 0/24 19/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 10/24 
agent 4: 0/24 */24 12/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 */24 10/24 
agent 4: 0/24 2/24 17/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 4/24 
agent 2: */24 12/24 0/24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 1/24 0/24 19/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 4/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: 1/24 0/24 19/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 */24 4/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 1/24 0/24 19/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 1/24 4/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 1/24 0/24 19/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 1/24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 */24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 */24 0/24 4/24 
agent 2: */24 19/24 0/24 4/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: */24 2/24 12/24 6/24 
agent 4: 0/24 */24 12/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 1/24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 */24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 */24 */24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 2/24 17/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 2/24 17/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 19/24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 3/24 0/24 */24 
agent 3: */24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 3/24 0/24 */24 
agent 3: */24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 3/24 0/24 10/24 
agent 3: */24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: */24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 11/24 3/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 2/24 18/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 */24 0/24 */24 
agent 4: 0/24 2/24 18/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 6/24 3/24 
agent 3: 8/24 */24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: */24 7/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 6/24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 18/24 4/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 1/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: */24 7/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 9/24 2/24 
agent 2: */24 0/24 3/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: */24 */24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 3/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 9/24 2/24 
agent 2: */24 0/24 3/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: */24 0/24 3/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: */24 0/24 3/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 11/24 */24 3/24 
agent 4: 0/24 11/24 */24 3/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 4/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 1/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 1/24 */24 
agent 4: 0/24 11/24 1/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 0/24 4/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: */24 9/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 */24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 9/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 9/24 */24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 9/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 9/24 0/24 4/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: */24 19/24 0/24 4/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 9/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: */24 19/24 0/24 4/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 11/24 9/24 0/24 4/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 */24 0/24 4/24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 */24 
agent 4: 0/24 1/24 18/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 */24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 */24 0/24 4/24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 0/24 4/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 1/24 4/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 9/24 0/24 4/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 19/24 */24 4/24 
agent 4: 0/24 1/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 1/24 4/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 4/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 0/24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 4/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: */24 0/24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 17/24 6/24 */24 
agent 4: 0/24 */24 4/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 17/24 6/24 */24 
agent 4: 0/24 */24 4/24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 1/24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 17/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 1/24 4/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 6/24 1/24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 17/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 */24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 */24 17/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: 8/24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: */24 0/24 3/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 3/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 1/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 */24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 */24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 0/24 4/24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 18/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 6/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: */24 */24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: */24 9/24 4/24 0/24 
agent 3: */24 0/24 10/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: 11/24 9/24 4/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 11/24 */24 4/24 */24 
agent 3: */24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 11/24 9/24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 6/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 11/24 */24 4/24 */24 
agent 3: */24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 11/24 */24 4/24 */24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 4/24 0/24 
agent 2: 11/24 */24 4/24 */24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: 11/24 */24 4/24 */24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 2/24 0/24 
agent 2: 11/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 2/24 9/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 2/24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 2/24 0/24 
agent 2: 11/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 */24 */24 */24 
agent 3: 2/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 */24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 */24 
agent 3: 2/24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 */24 */24 
agent 3: 2/24 8/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 9/24 2/24 
agent 2: */24 */24 3/24 */24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 */24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 8/24 */24 3/24 
agent 3: */24 6/24 */24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 8/24 5/24 */24 
agent 3: 8/24 1/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 3/24 */24 8/24 
agent 4: 0/24 3/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 9/24 2/24 
agent 2: */24 */24 3/24 */24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 */24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: */24 */24 3/24 */24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 */24 2/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 */24 2/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: */24 */24 */24 4/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 0/24 3/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 3/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 3/24 
agent 2: 11/24 */24 8/24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 0/24 3/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 3/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 3/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: 2/24 11/24 9/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 */24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 10/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 */24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 2/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 14/24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 9/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 */24 18/24 5/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 11/24 */24 5/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 8/24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 14/24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 14/24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 */24 12/24 9/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 10/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 */24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 2/24 18/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 */24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 21/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 11/24 3/24 */24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 11/24 3/24 */24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 */24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 23/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 */24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 11/24 */24 */24 0/24 
agent 3: 2/24 */24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 11/24 */24 */24 0/24 
agent 3: 2/24 */24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 11/24 */24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 2/24 0/24 
agent 2: 11/24 */24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 10/24 0/24 
agent 3: 2/24 */24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 */24 */24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 */24 */24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 10/24 
agent 4: 0/24 */24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 0/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 10/24 
agent 4: 0/24 */24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 0/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 0/24 11/24 
agent 4: 0/24 */24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 11/24 9/24 0/24 4/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 0/24 11/24 
agent 4: 0/24 */24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 2/24 11/24 0/24 11/24 
agent 4: 0/24 */24 */24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: 11/24 0/24 */24 */24 
agent 3: 2/24 11/24 0/24 11/24 
agent 4: 0/24 */24 */24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: 11/24 0/24 2/24 */24 
agent 3: */24 11/24 3/24 */24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: 11/24 0/24 2/24 */24 
agent 3: */24 11/24 3/24 */24 
agent 4: 0/24 */24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: */24 11/24 3/24 */24 
agent 4: 0/24 */24 */24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 11/24 3/24 8/24 
agent 4: 0/24 */24 */24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: */24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 1/24 
agent 2: 8/24 */24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 1/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 */24 */24 2/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 8/24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 8/24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 11/24 3/24 8/24 
agent 4: 0/24 */24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 3/24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 8/24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 11/24 3/24 8/24 
agent 4: 0/24 */24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 11/24 3/24 8/24 
agent 4: 0/24 */24 */24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 11/24 3/24 8/24 
agent 4: 0/24 11/24 11/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 11/24 */24 5/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 11/24 */24 2/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 */24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 11/24 */24 5/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 11/24 */24 5/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 11/24 */24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 4/24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 */24 6/24 5/24 
agent 4: */24 11/24 0/24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 11/24 */24 5/24 
agent 4: 0/24 11/24 0/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 */24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 11/24 */24 3/24 
agent 4: 0/24 11/24 */24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 10/24 5/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 11/24 */24 3/24 
agent 4: 0/24 11/24 */24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 11/24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 2/24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 11/24 8/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 2/24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 1/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 1/24 */24 
agent 4: 0/24 11/24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: 2/24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 */24 1/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 1/24 */24 
agent 4: 0/24 11/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 1/24 */24 
agent 4: 0/24 11/24 1/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: */24 0/24 12/24 */24 
agent 3: */24 11/24 1/24 */24 
agent 4: 0/24 11/24 1/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 10/24 1/24 
agent 2: 8/24 */24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 1/24 
agent 2: */24 0/24 12/24 1/24 
agent 3: */24 */24 1/24 */24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 1/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 10/24 1/24 
agent 2: 8/24 */24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 1/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 1/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 6/24 */24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 1/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 14/24 9/24 1/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 6/24 */24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 6/24 5/24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 6/24 5/24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 5/24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 0/24 5/24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 8/24 0/24 5/24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 8/24 0/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 5/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 0/24 2/24 */24 
agent 3: */24 7/24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 5/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 0/24 5/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 0/24 5/24 
agent 2: 0/24 19/24 0/24 5/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 5/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 19/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 19/24 0/24 5/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 5/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 1/24 5/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 1/24 5/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 19/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 19/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 19/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 6/24 */24 
agent 4: 0/24 0/24 17/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 17/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 */24 
agent 2: 8/24 */24 0/24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 6/24 */24 
agent 4: 0/24 0/24 17/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 5/24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 5/24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 5/24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 8/24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: 2/24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 3/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 11/24 1/24 10/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 8/24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: 2/24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: */24 0/24 2/24 */24 
agent 3: 2/24 11/24 3/24 8/24 
agent 4: 0/24 11/24 11/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 11/24 3/24 8/24 
agent 4: 0/24 11/24 11/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 3/24 
agent 2: 11/24 */24 2/24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: */24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 3/24 8/24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 */24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 0/24 10/24 3/24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 10/24 3/24 
agent 3: */24 11/24 */24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: */24 0/24 12/24 */24 
agent 3: 2/24 11/24 1/24 */24 
agent 4: 0/24 11/24 1/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 3/24 
agent 2: 11/24 0/24 10/24 3/24 
agent 3: */24 11/24 */24 5/24 
agent 4: 0/24 11/24 0/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 */24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: */24 */24 8/24 3/24 
agent 4: 0/24 */24 */24 3/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 1/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 14/24 9/24 1/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 8/24 3/24 
agent 2: 11/24 0/24 10/24 3/24 
agent 3: */24 11/24 */24 5/24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 2/24 8/24 3/24 
agent 2: 11/24 0/24 10/24 3/24 
agent 3: */24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 0/24 10/24 3/24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 */24 0/24 3/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: */24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 1/24 
agent 2: */24 0/24 12/24 1/24 
agent 3: 2/24 11/24 1/24 */24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 6/24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 */24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: */24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: */24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 11/24 3/24 8/24 
agent 4: 0/24 11/24 11/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 */24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 */24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 14/24 9/24 1/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 */24 */24 3/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 */24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 1/24 
agent 2: */24 0/24 12/24 1/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 1/24 
agent 2: */24 0/24 12/24 1/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 */24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 1/24 
agent 2: 11/24 0/24 12/24 1/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 */24 1/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 10/24 1/24 
agent 2: 11/24 0/24 12/24 1/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 */24 1/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 0/24 12/24 1/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 11/24 1/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 10/24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 14/24 9/24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 */24 */24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 0/24 12/24 */24 
agent 3: 2/24 11/24 1/24 */24 
agent 4: 0/24 11/24 1/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 11/24 1/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 */24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 11/24 1/24 10/24 
agent 4: 0/24 11/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 10/24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 14/24 9/24 1/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 6/24 5/24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 */24 */24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 5/24 
agent 2: 8/24 */24 0/24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 0/24 17/24 7/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 10/24 5/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 */24 */24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 0/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 11/24 */24 6/24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 3/24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 11/24 10/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 11/24 */24 6/24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 11/24 */24 6/24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 0/24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 11/24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 2/24 */24 0/24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 2/24 */24 0/24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 0/24 
agent 2: 11/24 0/24 */24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 2/24 */24 0/24 
agent 2: 11/24 0/24 */24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 */24 0/24 
agent 2: 11/24 0/24 */24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 0/24 */24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 11/24 0/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 0/24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 3/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 11/24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 */24 6/24 
agent 3: 2/24 10/24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 11/24 0/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 */24 6/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 14/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 11/24 */24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 11/24 */24 6/24 
agent 4: 0/24 11/24 7/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 2/24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 2/24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 2/24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 1/24 10/24 
agent 4: 0/24 11/24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 9/24 
agent 4: 0/24 */24 */24 9/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 4/24 7/24 
agent 4: */24 11/24 4/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 0/24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 1/24 10/24 
agent 4: 0/24 11/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 11/24 1/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 11/24 1/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 11/24 10/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: */24 0/24 */24 3/24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 10/24 
agent 4: */24 11/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 2/24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 11/24 10/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 2/24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 11/24 10/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 2/24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 11/24 8/24 */24 
agent 4: 0/24 11/24 10/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 2/24 6/24 */24 
agent 2: */24 0/24 0/24 */24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 3/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 11/24 8/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 8/24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 8/24 3/24 
agent 4: */24 11/24 8/24 3/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 1/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 1/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 2/24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 8/24 3/24 
agent 4: */24 11/24 8/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 8/24 3/24 
agent 4: 2/24 11/24 8/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 0/24 0/24 14/24 
agent 3: */24 */24 8/24 3/24 
agent 4: 2/24 11/24 8/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: */24 */24 8/24 3/24 
agent 4: 2/24 11/24 8/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: 10/24 0/24 */24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 11/24 */24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: */24 11/24 8/24 */24 
agent 4: */24 11/24 8/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 11/24 */24 2/24 
agent 4: 2/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: 10/24 0/24 */24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 11/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: 10/24 0/24 8/24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 11/24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: 10/24 0/24 8/24 6/24 
agent 3: */24 */24 5/24 6/24 
agent 4: */24 11/24 5/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 6/24 
agent 2: 10/24 0/24 8/24 6/24 
agent 3: */24 */24 5/24 6/24 
agent 4: 2/24 11/24 5/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 6/24 6/24 
agent 2: 10/24 0/24 8/24 6/24 
agent 3: */24 11/24 */24 6/24 
agent 4: */24 11/24 */24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 14/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 4/24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 */24 6/24 5/24 
agent 4: 2/24 11/24 0/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 6/24 6/24 
agent 2: 10/24 0/24 8/24 6/24 
agent 3: */24 11/24 */24 6/24 
agent 4: */24 11/24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 6/24 6/24 
agent 2: 10/24 0/24 8/24 6/24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 2/24 11/24 5/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 6/24 6/24 
agent 2: 10/24 */24 6/24 6/24 
agent 3: */24 */24 6/24 6/24 
agent 4: */24 */24 6/24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 5/24 6/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 4/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 6/24 */24 
agent 4: 2/24 11/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 6/24 6/24 
agent 2: 10/24 */24 6/24 6/24 
agent 3: */24 */24 6/24 6/24 
agent 4: */24 */24 6/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 6/24 6/24 
agent 2: 10/24 2/24 6/24 6/24 
agent 3: */24 */24 6/24 6/24 
agent 4: */24 */24 6/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 6/24 6/24 
agent 2: 10/24 2/24 6/24 6/24 
agent 3: 2/24 10/24 6/24 6/24 
agent 4: 2/24 10/24 6/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: */24 */24 8/24 4/24 
agent 4: */24 */24 8/24 4/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 6/24 6/24 
agent 4: 0/24 10/24 */24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 4/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 4/24 
agent 4: 2/24 10/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 6/24 6/24 
agent 4: 0/24 10/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 6/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 5/24 6/24 
agent 2: */24 */24 5/24 6/24 
agent 3: */24 */24 6/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 5/24 6/24 
agent 2: 8/24 */24 5/24 6/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 5/24 */24 6/24 
agent 4: 0/24 */24 11/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 10/24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 6/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 5/24 6/24 
agent 2: 8/24 */24 5/24 6/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 5/24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 5/24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 4/24 6/24 
agent 4: 6/24 0/24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 */24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 6/24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 8/24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: */24 12/24 */24 6/24 
agent 4: */24 0/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 */24 8/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: */24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 */24 8/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 */24 8/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 */24 8/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: 0/24 */24 11/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 10/24 8/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: 0/24 */24 11/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: 0/24 7/24 11/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: */24 6/24 */24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: */24 5/24 11/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 2/24 */24 
agent 3: 2/24 5/24 11/24 6/24 
agent 4: 0/24 7/24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: */24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: */24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 */24 
agent 4: 0/24 8/24 10/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: */24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: */24 */24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: */24 8/24 */24 6/24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 8/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: */24 8/24 */24 6/24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 8/24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: */24 8/24 */24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: */24 8/24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 6/24 
agent 2: 6/24 */24 */24 6/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: */24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 10/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > b > d
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 6/24 
agent 2: 6/24 */24 */24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 4/24 6/24 
agent 4: 6/24 0/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > c > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 4/24 6/24 
agent 4: 6/24 0/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 4/24 6/24 
agent 4: 6/24 0/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 4/24 6/24 
agent 2: 6/24 8/24 4/24 6/24 
agent 3: 6/24 8/24 4/24 6/24 
agent 4: 6/24 0/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 6/24 
agent 2: */24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 10/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 6/24 
agent 2: 6/24 */24 */24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: */24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 */24 5/24 
agent 4: 6/24 0/24 4/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 6/24 
agent 2: 6/24 */24 */24 6/24 
agent 3: 6/24 */24 */24 6/24 
agent 4: */24 */24 */24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 6/24 8/24 0/24 */24 
agent 4: */24 0/24 */24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 6/24 
agent 2: */24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 10/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 4/24 6/24 
agent 2: 5/24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 10/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 4/24 6/24 
agent 2: 5/24 8/24 5/24 6/24 
agent 3: 5/24 8/24 5/24 6/24 
agent 4: 0/24 8/24 10/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 6/24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 0/24 
agent 2: */24 8/24 */24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 5/24 8/24 0/24 */24 
agent 4: 0/24 8/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 0/24 6/24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 4/24 0/24 6/24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 4/24 0/24 6/24 
agent 2: 5/24 5/24 8/24 6/24 
agent 3: 5/24 5/24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 14/24 4/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 6/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 5/24 */24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: 0/24 7/24 11/24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 5/24 0/24 8/24 */24 
agent 4: 0/24 12/24 8/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 4/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 */24 6/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 4/24 6/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 10/24 6/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 4/24 6/24 0/24 
agent 2: 5/24 5/24 6/24 8/24 
agent 3: 5/24 5/24 6/24 8/24 
agent 4: 0/24 10/24 6/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: */24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 5/24 6/24 2/24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 6/24 8/24 
agent 4: */24 */24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 5/24 0/24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 10/24 6/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 6/24 8/24 
agent 4: */24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 */24 6/24 8/24 
agent 4: */24 */24 6/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 5/24 6/24 2/24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: */24 */24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 5/24 6/24 2/24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 11/24 5/24 6/24 2/24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 5/24 6/24 2/24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 0/24 
agent 2: */24 */24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: */24 5/24 */24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: 0/24 7/24 11/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 0/24 */24 2/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 0/24 0/24 
agent 2: */24 */24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: */24 */24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: */24 5/24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 3/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 0/24 
agent 2: */24 5/24 6/24 8/24 
agent 3: */24 */24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 3/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: */24 5/24 */24 */24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 3/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: */24 */24 */24 6/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 4/24 8/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 0/24 8/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 3/24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 4/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 5/24 5/24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: */24 5/24 */24 6/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 6/24 8/24 */24 
agent 3: */24 5/24 8/24 */24 
agent 4: */24 */24 8/24 10/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 5/24 */24 8/24 */24 
agent 3: 5/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: */24 5/24 */24 6/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 3/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: */24 5/24 */24 6/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 3/24 */24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: */24 5/24 */24 6/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 3/24 11/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 */24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: */24 3/24 11/24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 3/24 */24 8/24 
agent 4: 0/24 3/24 11/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 */24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: */24 3/24 11/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 */24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 2/24 3/24 11/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 10/24 2/24 */24 */24 
agent 3: 2/24 */24 11/24 */24 
agent 4: */24 2/24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 3/24 */24 8/24 
agent 4: 0/24 3/24 11/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 3/24 11/24 8/24 
agent 4: 0/24 3/24 11/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 2/24 6/24 
agent 3: 2/24 3/24 11/24 8/24 
agent 4: 0/24 3/24 11/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: */24 */24 2/24 1/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 11/24 */24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 */24 */24 
agent 3: 2/24 8/24 11/24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 */24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 */24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 11/24 */24 
agent 4: 0/24 3/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 11/24 10/24 
agent 4: 0/24 3/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 11/24 10/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 11/24 10/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 11/24 10/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 11/24 */24 
agent 4: 0/24 2/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 */24 0/24 18/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 3/24 9/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 11/24 10/24 
agent 4: 0/24 3/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 */24 0/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 6/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 6/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 6/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 6/24 8/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 6/24 8/24 */24 
agent 3: */24 */24 8/24 10/24 
agent 4: 0/24 6/24 8/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 6/24 8/24 */24 
agent 3: */24 */24 8/24 10/24 
agent 4: 0/24 6/24 8/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: */24 */24 8/24 10/24 
agent 4: 0/24 6/24 8/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 3/24 11/24 10/24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 0/24 */24 8/24 10/24 
agent 4: 0/24 6/24 8/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 3/24 11/24 10/24 
agent 4: 0/24 3/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 11/24 10/24 
agent 4: 0/24 3/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 11/24 10/24 
agent 4: 0/24 3/24 11/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 2/24 0/24 
agent 3: 0/24 1/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 11/24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 11/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 2/24 0/24 
agent 3: 0/24 1/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 2/24 0/24 
agent 3: 0/24 1/24 11/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 2/24 0/24 
agent 3: 0/24 1/24 11/24 12/24 
agent 4: 0/24 1/24 11/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: */24 */24 2/24 1/24 
agent 3: */24 1/24 11/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 */24 0/24 
agent 3: 0/24 */24 11/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 11/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 1/24 11/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: */24 */24 2/24 1/24 
agent 3: */24 1/24 11/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: */24 */24 2/24 1/24 
agent 3: */24 1/24 11/24 */24 
agent 4: 0/24 */24 11/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 1/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 8/24 */24 3/24 
agent 3: */24 6/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 11/24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 1/24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 1/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 8/24 */24 3/24 
agent 3: */24 6/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 8/24 2/24 3/24 
agent 3: 2/24 6/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 8/24 2/24 3/24 
agent 3: 2/24 6/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 8/24 3/24 
agent 2: 11/24 */24 8/24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 */24 0/24 3/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 2/24 */24 
agent 3: 2/24 8/24 11/24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: */24 */24 2/24 1/24 
agent 3: 2/24 1/24 11/24 */24 
agent 4: 0/24 */24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 8/24 3/24 
agent 2: 11/24 */24 8/24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 2/24 8/24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 2/24 8/24 3/24 
agent 3: 2/24 10/24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 */24 2/24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 2/24 3/24 */24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 10/24 */24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 1/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 */24 2/24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 */24 2/24 */24 
agent 3: 2/24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 3/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 */24 11/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: */24 8/24 2/24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 */24 */24 4/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: 2/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 0/24 3/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 */24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 */24 11/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: 2/24 2/24 */24 9/24 
agent 4: 0/24 10/24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: 2/24 2/24 11/24 9/24 
agent 4: 0/24 10/24 11/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 2/24 2/24 9/24 
agent 3: 2/24 2/24 11/24 9/24 
agent 4: 0/24 10/24 11/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 2/24 9/24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 2/24 8/24 9/24 
agent 3: */24 2/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 */24 2/24 */24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 11/24 5/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 11/24 */24 
agent 4: 0/24 2/24 11/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 0/24 2/24 */24 
agent 3: 2/24 7/24 11/24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 10/24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 2/24 9/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: 2/24 2/24 9/24 */24 
agent 4: 0/24 10/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: 2/24 2/24 9/24 11/24 
agent 4: 0/24 10/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 */24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 2/24 9/24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 */24 */24 14/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 10/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 2/24 9/24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 2/24 9/24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 6/24 4/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 4/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 6/24 */24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 4/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 5/24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 18/24 4/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 5/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 5/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 5/24 19/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: 2/24 2/24 9/24 11/24 
agent 4: 0/24 10/24 3/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: 2/24 2/24 9/24 11/24 
agent 4: 0/24 10/24 3/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: 2/24 2/24 9/24 11/24 
agent 4: 0/24 10/24 3/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 9/24 2/24 
agent 3: 2/24 2/24 9/24 11/24 
agent 4: 0/24 10/24 3/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 9/24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: 11/24 */24 4/24 2/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 10/24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 2/24 9/24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 5/24 2/24 9/24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 5/24 */24 4/24 8/24 
agent 3: */24 0/24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 5/24 */24 4/24 8/24 
agent 3: */24 0/24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 5/24 7/24 4/24 8/24 
agent 3: */24 0/24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 5/24 */24 0/24 
agent 2: 5/24 7/24 4/24 8/24 
agent 3: */24 0/24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 5/24 0/24 */24 
agent 2: */24 */24 8/24 4/24 
agent 3: */24 0/24 8/24 */24 
agent 4: 0/24 12/24 8/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: 11/24 7/24 4/24 2/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 5/24 0/24 */24 
agent 2: */24 */24 8/24 4/24 
agent 3: */24 0/24 8/24 */24 
agent 4: 0/24 12/24 8/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 5/24 0/24 */24 
agent 2: */24 7/24 8/24 4/24 
agent 3: */24 0/24 8/24 */24 
agent 4: 0/24 12/24 8/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 5/24 0/24 */24 
agent 2: 5/24 7/24 8/24 4/24 
agent 3: */24 0/24 8/24 */24 
agent 4: 0/24 12/24 8/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 7/24 2/24 4/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 11/24 5/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 5/24 2/24 8/24 9/24 
agent 3: */24 2/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 7/24 2/24 4/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 11/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 7/24 2/24 4/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 8/24 11/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 11/24 */24 
agent 4: 0/24 0/24 11/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 0/24 11/24 11/24 
agent 4: 0/24 8/24 11/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 11/24 */24 
agent 4: 0/24 0/24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 */24 11/24 */24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 2/24 3/24 
agent 3: 0/24 */24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 7/24 2/24 3/24 
agent 3: 0/24 8/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 7/24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 2/24 */24 
agent 3: 0/24 10/24 11/24 */24 
agent 4: 0/24 2/24 11/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 17/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 8/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 11/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 7/24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 6/24 7/24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: 6/24 7/24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 4/24 0/24 2/24 
agent 2: */24 3/24 8/24 */24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 2/24 
agent 2: */24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 6/24 3/24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 4/24 0/24 2/24 
agent 2: */24 3/24 8/24 */24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: */24 3/24 8/24 */24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: 6/24 3/24 8/24 7/24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 2/24 
agent 2: */24 8/24 3/24 */24 
agent 3: 0/24 8/24 */24 1/24 
agent 4: 0/24 8/24 2/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: */24 3/24 */24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 */24 7/24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 6/24 */24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 2/24 
agent 2: */24 8/24 3/24 */24 
agent 3: 0/24 8/24 */24 1/24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: */24 8/24 3/24 */24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: 6/24 8/24 3/24 7/24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 4/24 2/24 
agent 2: 6/24 12/24 */24 */24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: */24 0/24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 3/24 7/24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 2/24 
agent 2: 6/24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 1/24 
agent 3: 5/24 8/24 3/24 */24 
agent 4: 0/24 8/24 2/24 14/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 4/24 2/24 
agent 2: 6/24 12/24 */24 */24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: */24 0/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 */24 */24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: */24 0/24 1/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 2/24 */24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: 6/24 0/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: 6/24 0/24 1/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 1/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 6/24 */24 2/24 12/24 
agent 4: 6/24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 1/24 
agent 2: 6/24 */24 */24 1/24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 2/24 */24 */24 
agent 4: 6/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: 6/24 0/24 */24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: */24 0/24 */24 */24 
agent 4: */24 0/24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: */24 0/24 16/24 */24 
agent 4: */24 0/24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: */24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: */24 0/24 16/24 2/24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: */24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: */24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: 6/24 8/24 7/24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 7/24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 6/24 8/24 3/24 */24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 7/24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 7/24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 7/24 3/24 
agent 3: 0/24 */24 10/24 4/24 
agent 4: 0/24 */24 0/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 7/24 3/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 10/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 10/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 */24 */24 4/24 
agent 3: */24 10/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 10/24 0/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 3/24 7/24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 3/24 7/24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 6/24 8/24 3/24 7/24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 */24 2/24 */24 
agent 3: 2/24 10/24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 6/24 8/24 3/24 7/24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 6/24 8/24 3/24 7/24 
agent 3: 0/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 6/24 3/24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 2/24 
agent 2: 6/24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 6/24 3/24 8/24 */24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 6/24 3/24 8/24 7/24 
agent 3: 0/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: */24 3/24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 2/24 7/24 
agent 3: 0/24 10/24 11/24 */24 
agent 4: 0/24 2/24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: */24 3/24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 6/24 3/24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 6/24 3/24 7/24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 2/24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 11/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 2/24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 7/24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: 6/24 */24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: 6/24 3/24 */24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: 6/24 */24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: 6/24 7/24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 7/24 */24 3/24 
agent 3: 0/24 */24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 7/24 */24 3/24 
agent 3: 0/24 */24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 7/24 2/24 3/24 
agent 3: 0/24 8/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 2/24 7/24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 8/24 11/24 */24 
agent 4: 0/24 0/24 11/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 8/24 11/24 */24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 8/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 8/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 10/24 11/24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 7/24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 10/24 11/24 */24 
agent 4: 0/24 2/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 10/24 11/24 3/24 
agent 4: 0/24 2/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 10/24 11/24 3/24 
agent 4: 0/24 2/24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 10/24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 2/24 7/24 
agent 3: 0/24 10/24 11/24 */24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 11/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 2/24 7/24 
agent 3: 0/24 10/24 11/24 */24 
agent 4: 0/24 2/24 11/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 17/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 17/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 17/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 2/24 7/24 
agent 3: 0/24 10/24 11/24 3/24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > c > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 2/24 7/24 
agent 3: 0/24 10/24 11/24 3/24 
agent 4: 0/24 2/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 2/24 7/24 
agent 3: 0/24 10/24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 2/24 */24 
agent 3: 2/24 8/24 11/24 3/24 
agent 4: 0/24 2/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 2/24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 11/24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 11/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 2/24 */24 
agent 3: 2/24 8/24 11/24 3/24 
agent 4: 0/24 2/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 2/24 8/24 
agent 3: 2/24 8/24 11/24 3/24 
agent 4: 0/24 2/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 2/24 8/24 
agent 3: 2/24 8/24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 1/24 
agent 3: 5/24 3/24 8/24 8/24 
agent 4: 0/24 2/24 8/24 14/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 8/24 5/24 3/24 
agent 3: 8/24 1/24 5/24 */24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 0/24 2/24 */24 
agent 3: 2/24 7/24 11/24 4/24 
agent 4: 0/24 */24 11/24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 11/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 1/24 
agent 3: 5/24 3/24 8/24 8/24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: */24 */24 8/24 1/24 
agent 3: 5/24 3/24 8/24 8/24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 1/24 
agent 2: */24 8/24 */24 1/24 
agent 3: 5/24 8/24 3/24 */24 
agent 4: 0/24 8/24 2/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 1/24 
agent 2: */24 8/24 */24 1/24 
agent 3: 5/24 8/24 3/24 */24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 1/24 
agent 2: */24 8/24 */24 1/24 
agent 3: 5/24 8/24 3/24 8/24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 1/24 
agent 2: */24 */24 */24 1/24 
agent 3: 2/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 2/24 3/24 8/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 1/24 
agent 2: */24 */24 */24 1/24 
agent 3: 2/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 1/24 
agent 2: 11/24 */24 */24 1/24 
agent 3: 2/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 1/24 
agent 2: */24 */24 2/24 1/24 
agent 3: 2/24 1/24 11/24 */24 
agent 4: 0/24 */24 11/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 6/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: 2/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 0/24 12/24 1/24 
agent 3: 2/24 11/24 1/24 */24 
agent 4: 0/24 11/24 1/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 10/24 0/24 
agent 3: 0/24 10/24 */24 12/24 
agent 4: 0/24 10/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 0/24 1/24 
agent 2: */24 */24 2/24 1/24 
agent 3: 2/24 1/24 11/24 */24 
agent 4: 0/24 */24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 */24 2/24 1/24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: 0/24 */24 11/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: 11/24 */24 2/24 1/24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: 0/24 */24 11/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: 11/24 10/24 2/24 1/24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: 0/24 1/24 11/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 */24 1/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: */24 10/24 8/24 1/24 
agent 3: 5/24 3/24 8/24 8/24 
agent 4: 0/24 2/24 8/24 14/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: 11/24 12/24 0/24 1/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 */24 2/24 6/24 
agent 3: 2/24 3/24 11/24 8/24 
agent 4: 0/24 3/24 11/24 10/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 8/24 5/24 3/24 
agent 3: 8/24 1/24 5/24 10/24 
agent 4: 0/24 */24 14/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 1/24 11/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 */24 1/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 */24 1/24 */24 
agent 3: 2/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: */24 0/24 */24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 */24 6/24 */24 
agent 3: 2/24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: 2/24 6/24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: */24 0/24 */24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 0/24 */24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 */24 
agent 2: 11/24 0/24 2/24 */24 
agent 3: 2/24 7/24 11/24 4/24 
agent 4: 0/24 */24 11/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 5/24 */24 0/24 
agent 2: 5/24 7/24 4/24 8/24 
agent 3: */24 0/24 11/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: 11/24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 */24 2/24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 */24 */24 */24 
agent 3: 2/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 0/24 */24 2/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 0/24 */24 
agent 2: 11/24 0/24 2/24 */24 
agent 3: 2/24 7/24 11/24 4/24 
agent 4: 0/24 */24 11/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 0/24 */24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 7/24 11/24 4/24 
agent 4: 0/24 9/24 11/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 8/24 0/24 5/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 7/24 11/24 4/24 
agent 4: 0/24 9/24 11/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 5/24 
agent 2: 0/24 19/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 6/24 5/24 
agent 2: 11/24 */24 0/24 */24 
agent 3: 2/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 5/24 0/24 */24 
agent 2: 5/24 7/24 8/24 4/24 
agent 3: */24 0/24 8/24 11/24 
agent 4: 0/24 12/24 8/24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 */24 2/24 */24 
agent 3: 2/24 5/24 11/24 6/24 
agent 4: 0/24 7/24 11/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 2/24 10/24 
agent 3: 0/24 */24 11/24 4/24 
agent 4: 0/24 */24 11/24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 7/24 */24 4/24 
agent 4: 0/24 9/24 11/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 5/24 
agent 2: 0/24 19/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 0/24 5/24 
agent 2: 0/24 19/24 0/24 5/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 5/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 19/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 */24 0/24 */24 
agent 2: 6/24 0/24 8/24 10/24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: 0/24 */24 8/24 5/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 2/24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 5/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 19/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 1/24 5/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 19/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 17/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 */24 
agent 2: 6/24 8/24 0/24 10/24 
agent 3: 1/24 8/24 11/24 4/24 
agent 4: 0/24 8/24 */24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 19/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 17/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 6/24 5/24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 2/24 */24 */24 
agent 4: 6/24 0/24 */24 7/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 6/24 5/24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 6/24 5/24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 1/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 8/24 5/24 5/24 
agent 3: */24 */24 0/24 10/24 
agent 4: */24 0/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 */24 1/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 1/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 10/24 1/24 
agent 2: 11/24 */24 */24 1/24 
agent 3: 2/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: */24 11/24 */24 2/24 
agent 4: 2/24 11/24 3/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 1/24 
agent 2: 6/24 */24 9/24 1/24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 10/24 1/24 
agent 2: 11/24 */24 */24 1/24 
agent 3: 2/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 */24 */24 1/24 
agent 3: 2/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 */24 0/24 */24 
agent 3: 2/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 */24 0/24 */24 
agent 3: 2/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 */24 0/24 11/24 
agent 3: 2/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 4/24 
agent 3: */24 8/24 0/24 11/24 
agent 4: 0/24 8/24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 2/24 */24 */24 
agent 3: */24 */24 6/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 4/24 
agent 3: */24 8/24 0/24 11/24 
agent 4: 0/24 8/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 4/24 
agent 3: 5/24 8/24 0/24 11/24 
agent 4: 0/24 8/24 12/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 5/24 
agent 2: */24 8/24 */24 4/24 
agent 3: 5/24 8/24 0/24 11/24 
agent 4: 0/24 8/24 12/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 5/24 
agent 2: 6/24 8/24 5/24 5/24 
agent 3: */24 */24 0/24 10/24 
agent 4: */24 0/24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 5/24 0/24 5/24 
agent 2: 5/24 7/24 8/24 4/24 
agent 3: */24 0/24 8/24 11/24 
agent 4: 0/24 12/24 8/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 5/24 8/24 */24 */24 
agent 4: 0/24 8/24 10/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 5/24 8/24 0/24 */24 
agent 4: 0/24 8/24 12/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 5/24 0/24 5/24 
agent 2: 5/24 7/24 8/24 4/24 
agent 3: */24 0/24 8/24 11/24 
agent 4: 0/24 12/24 8/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 5/24 0/24 5/24 
agent 2: 5/24 7/24 8/24 4/24 
agent 3: 5/24 0/24 8/24 11/24 
agent 4: 0/24 12/24 8/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 */24 0/24 5/24 
agent 2: 6/24 0/24 8/24 10/24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: 0/24 */24 8/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 5/24 
agent 2: */24 8/24 */24 4/24 
agent 3: 5/24 8/24 0/24 11/24 
agent 4: 0/24 8/24 12/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 5/24 */24 0/24 
agent 2: 5/24 7/24 4/24 8/24 
agent 3: */24 0/24 11/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 5/24 */24 8/24 */24 
agent 4: 0/24 10/24 8/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 */24 0/24 5/24 
agent 2: 6/24 0/24 8/24 10/24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: 0/24 */24 8/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 2/24 0/24 5/24 
agent 2: 6/24 0/24 8/24 10/24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: 0/24 11/24 8/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 5/24 
agent 2: 6/24 8/24 0/24 10/24 
agent 3: 1/24 8/24 11/24 4/24 
agent 4: 0/24 8/24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 2/24 */24 0/24 
agent 2: 6/24 0/24 10/24 8/24 
agent 3: 1/24 11/24 4/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 */24 8/24 */24 
agent 4: */24 11/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 0/24 */24 5/24 
agent 2: 6/24 8/24 0/24 10/24 
agent 3: 1/24 8/24 11/24 4/24 
agent 4: 0/24 8/24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 2/24 5/24 
agent 2: 6/24 8/24 0/24 10/24 
agent 3: 1/24 8/24 11/24 4/24 
agent 4: 0/24 8/24 11/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 5/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 2/24 */24 */24 
agent 4: 6/24 0/24 */24 7/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 8/24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 2/24 5/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 2/24 */24 */24 
agent 4: 6/24 0/24 */24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 5/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 2/24 */24 */24 
agent 4: 6/24 0/24 11/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 5/24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 2/24 11/24 */24 
agent 4: 6/24 0/24 11/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 0/24 6/24 
agent 3: */24 */24 11/24 */24 
agent 4: */24 0/24 */24 7/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 0/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 6/24 */24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > c > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 0/24 6/24 
agent 3: */24 0/24 11/24 */24 
agent 4: */24 0/24 */24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 0/24 6/24 
agent 3: 6/24 0/24 11/24 */24 
agent 4: 6/24 0/24 */24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 0/24 6/24 
agent 3: 6/24 0/24 11/24 7/24 
agent 4: 6/24 0/24 11/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 2/24 4/24 
agent 2: 6/24 12/24 0/24 6/24 
agent 3: 6/24 0/24 11/24 7/24 
agent 4: 6/24 0/24 11/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 2/24 4/24 
agent 2: 6/24 8/24 0/24 */24 
agent 3: 0/24 8/24 11/24 */24 
agent 4: 0/24 8/24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 4/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 5/24 
agent 2: */24 */24 0/24 */24 
agent 3: 6/24 2/24 11/24 */24 
agent 4: 6/24 0/24 11/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 2/24 4/24 
agent 2: 6/24 8/24 0/24 */24 
agent 3: 0/24 8/24 11/24 */24 
agent 4: 0/24 8/24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 2/24 4/24 
agent 2: 6/24 8/24 0/24 10/24 
agent 3: 0/24 8/24 11/24 5/24 
agent 4: 0/24 8/24 11/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 4/24 
agent 2: 6/24 0/24 8/24 */24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 */24 0/24 10/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 4/24 
agent 2: 6/24 0/24 8/24 */24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 */24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 2/24 0/24 4/24 
agent 2: 6/24 0/24 8/24 10/24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 2/24 0/24 4/24 
agent 2: 6/24 0/24 8/24 10/24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 11/24 8/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 5/24 2/24 8/24 9/24 
agent 3: */24 2/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 2/24 */24 0/24 
agent 2: 6/24 0/24 */24 8/24 
agent 3: 0/24 11/24 5/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 */24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 2/24 */24 0/24 
agent 2: 6/24 0/24 */24 8/24 
agent 3: 0/24 11/24 5/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 2/24 4/24 0/24 
agent 2: 6/24 0/24 10/24 8/24 
agent 3: 0/24 11/24 5/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 2/24 4/24 0/24 
agent 2: 6/24 0/24 10/24 8/24 
agent 3: 0/24 11/24 5/24 8/24 
agent 4: 0/24 11/24 5/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 5/24 2/24 9/24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 10/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 10/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 7/24 */24 11/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 7/24 */24 11/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 */24 */24 11/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 1/24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 3/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 17/24 */24 0/24 
agent 4: 0/24 1/24 */24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 17/24 */24 0/24 
agent 4: 0/24 1/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 7/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 17/24 6/24 */24 
agent 4: 0/24 1/24 4/24 19/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 7/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 6/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 6/24 */24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 */24 */24 17/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 */24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 */24 17/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 6/24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: */24 2/24 6/24 12/24 
agent 4: */24 */24 6/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 17/24 6/24 */24 
agent 4: 0/24 1/24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: */24 2/24 6/24 12/24 
agent 4: */24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: */24 */24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: */24 */24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 11/24 */24 6/24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 */24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 11/24 */24 6/24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 0/24 
agent 2: */24 5/24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 0/24 */24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 7/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 */24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: 0/24 6/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 0/24 0/24 
agent 2: */24 5/24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: */24 5/24 0/24 8/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 3/24 12/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 6/24 */24 
agent 3: */24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 11/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 */24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: */24 5/24 0/24 8/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 0/24 8/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 5/24 0/24 18/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 */24 6/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 3/24 11/24 10/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 */24 0/24 8/24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 5/24 0/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 5/24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 5/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: */24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 5/24 0/24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: 1/24 */24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 */24 
agent 4: */24 0/24 */24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: */24 12/24 6/24 4/24 
agent 4: */24 0/24 5/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 5/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 7/24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 */24 17/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: */24 12/24 */24 6/24 
agent 4: */24 0/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 */24 6/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: */24 0/24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 5/24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 7/24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 */24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 */24 17/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 5/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 5/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 5/24 17/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 */24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 9/24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: */24 12/24 */24 6/24 
agent 4: */24 0/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 5/24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 5/24 6/24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 5/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 5/24 6/24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 9/24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 5/24 6/24 
agent 2: */24 0/24 */24 6/24 
agent 3: */24 */24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 5/24 6/24 
agent 2: */24 0/24 7/24 6/24 
agent 3: */24 */24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 5/24 6/24 
agent 2: 11/24 0/24 7/24 6/24 
agent 3: */24 */24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 5/24 6/24 
agent 2: 11/24 */24 5/24 6/24 
agent 3: */24 */24 6/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: 11/24 0/24 7/24 */24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 5/24 6/24 
agent 2: 11/24 0/24 7/24 6/24 
agent 3: */24 */24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 5/24 6/24 
agent 2: 11/24 0/24 7/24 6/24 
agent 3: */24 */24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 5/24 6/24 
agent 2: 11/24 0/24 7/24 6/24 
agent 3: */24 */24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 5/24 6/24 
agent 2: 11/24 0/24 7/24 6/24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 0/24 
agent 2: */24 0/24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 */24 */24 
agent 3: 2/24 11/24 0/24 */24 
agent 4: 0/24 11/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 */24 0/24 
agent 2: */24 0/24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: */24 0/24 5/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: */24 0/24 5/24 */24 
agent 3: */24 11/24 4/24 */24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 5/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: */24 */24 5/24 6/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 5/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: */24 */24 5/24 6/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 11/24 0/24 
agent 2: */24 */24 5/24 6/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: */24 */24 5/24 6/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 5/24 6/24 
agent 2: 11/24 */24 5/24 6/24 
agent 3: */24 */24 6/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 0/24 
agent 2: */24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 4/24 */24 
agent 4: 0/24 10/24 4/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 5/24 5/24 
agent 4: 6/24 0/24 4/24 14/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 8/24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 5/24 5/24 
agent 4: 6/24 0/24 4/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 10/24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: */24 */24 5/24 5/24 
agent 4: 6/24 0/24 4/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 10/24 0/24 
agent 2: 6/24 8/24 5/24 5/24 
agent 3: 6/24 8/24 5/24 5/24 
agent 4: 6/24 0/24 4/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 10/24 0/24 
agent 2: */24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 0/24 
agent 2: */24 */24 12/24 4/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 6/24 */24 
agent 3: 6/24 */24 5/24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 0/24 10/24 0/24 
agent 2: */24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 10/24 0/24 
agent 2: 5/24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 10/24 0/24 
agent 2: 5/24 8/24 5/24 6/24 
agent 3: 5/24 8/24 5/24 6/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 0/24 
agent 2: 5/24 5/24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 6/24 */24 
agent 3: */24 8/24 5/24 */24 
agent 4: */24 8/24 */24 10/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 5/24 8/24 */24 */24 
agent 3: 5/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 0/24 0/24 
agent 2: 5/24 5/24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 10/24 0/24 0/24 
agent 2: 5/24 5/24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 10/24 0/24 0/24 
agent 2: 5/24 5/24 8/24 6/24 
agent 3: 5/24 5/24 8/24 6/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 10/24 0/24 0/24 
agent 2: */24 5/24 6/24 8/24 
agent 3: */24 */24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 10/24 0/24 0/24 
agent 2: */24 5/24 6/24 8/24 
agent 3: */24 */24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 10/24 0/24 0/24 
agent 2: 5/24 5/24 6/24 8/24 
agent 3: */24 */24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 10/24 0/24 0/24 
agent 2: 5/24 5/24 6/24 8/24 
agent 3: 5/24 5/24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 */24 8/24 
agent 3: */24 5/24 */24 8/24 
agent 4: */24 */24 10/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 5/24 8/24 */24 */24 
agent 3: 5/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: 5/24 8/24 */24 */24 
agent 3: 5/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 */24 
agent 2: 5/24 */24 8/24 */24 
agent 3: 5/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 5/24 8/24 0/24 */24 
agent 4: 0/24 8/24 12/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 5/24 8/24 0/24 */24 
agent 4: 0/24 8/24 12/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 5/24 8/24 0/24 11/24 
agent 4: 0/24 8/24 12/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: 5/24 8/24 */24 */24 
agent 3: 5/24 8/24 0/24 11/24 
agent 4: 0/24 8/24 12/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 5/24 0/24 8/24 */24 
agent 4: 0/24 12/24 8/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: 5/24 8/24 2/24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 */24 
agent 2: */24 */24 0/24 11/24 
agent 3: */24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 8/24 */24 4/24 
agent 4: 4/24 8/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 5/24 0/24 8/24 */24 
agent 4: 0/24 12/24 8/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 5/24 0/24 8/24 11/24 
agent 4: 0/24 12/24 8/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 */24 0/24 */24 
agent 2: 5/24 */24 8/24 */24 
agent 3: 5/24 0/24 8/24 11/24 
agent 4: 0/24 12/24 8/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 5/24 0/24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 5/24 */24 8/24 */24 
agent 3: */24 2/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 11/24 
agent 3: */24 7/24 */24 4/24 
agent 4: 0/24 9/24 11/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 5/24 0/24 */24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 5/24 0/24 11/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: 5/24 0/24 11/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: */24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: 5/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 8/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 5/24 */24 3/24 8/24 
agent 4: 0/24 0/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 8/24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 5/24 */24 3/24 8/24 
agent 4: 0/24 0/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 8/24 2/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 5/24 8/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 8/24 2/24 0/24 
agent 2: 5/24 8/24 3/24 8/24 
agent 3: 5/24 8/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 8/24 3/24 */24 
agent 3: */24 6/24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 8/24 */24 0/24 
agent 2: */24 */24 3/24 */24 
agent 3: */24 */24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: 1/24 12/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: 5/24 2/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: 5/24 2/24 9/24 8/24 
agent 4: 0/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 */24 
agent 2: 5/24 */24 8/24 */24 
agent 3: */24 2/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 3/24 0/24 
agent 2: 11/24 2/24 9/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 10/24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 0/24 */24 
agent 2: 5/24 */24 8/24 */24 
agent 3: */24 2/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 */24 0/24 */24 
agent 2: 5/24 */24 8/24 */24 
agent 3: 5/24 2/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 */24 0/24 */24 
agent 2: 5/24 */24 8/24 */24 
agent 3: 5/24 2/24 8/24 9/24 
agent 4: 0/24 */24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 2/24 */24 9/24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 10/24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 2/24 */24 9/24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 10/24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 3/24 
agent 2: 11/24 2/24 2/24 9/24 
agent 3: */24 2/24 */24 9/24 
agent 4: 0/24 10/24 11/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 0/24 3/24 
agent 2: 11/24 2/24 2/24 9/24 
agent 3: */24 2/24 11/24 9/24 
agent 4: 0/24 10/24 11/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 2/24 2/24 9/24 
agent 3: 2/24 2/24 11/24 9/24 
agent 4: 0/24 10/24 11/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 9/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 10/24 */24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 */24 
agent 2: 5/24 2/24 8/24 9/24 
agent 3: 5/24 2/24 8/24 9/24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 */24 2/24 */24 
agent 3: */24 0/24 11/24 11/24 
agent 4: 0/24 8/24 11/24 5/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 11/24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 */24 11/24 
agent 3: 2/24 7/24 11/24 4/24 
agent 4: 0/24 9/24 11/24 4/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 10/24 2/24 */24 */24 
agent 3: 2/24 */24 11/24 3/24 
agent 4: */24 2/24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 9/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 10/24 */24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 9/24 2/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 10/24 3/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 9/24 2/24 
agent 3: 2/24 2/24 9/24 11/24 
agent 4: 0/24 10/24 3/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: 5/24 2/24 9/24 8/24 
agent 3: 5/24 2/24 9/24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 */24 2/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 6/24 */24 14/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 11/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 10/24 2/24 */24 */24 
agent 3: 2/24 */24 3/24 11/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 */24 2/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 */24 */24 2/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 */24 5/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 */24 */24 2/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 8/24 5/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 2/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 */24 
agent 2: 11/24 */24 2/24 */24 
agent 3: */24 0/24 11/24 11/24 
agent 4: 0/24 8/24 11/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 10/24 */24 4/24 */24 
agent 3: */24 6/24 5/24 11/24 
agent 4: */24 0/24 11/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: 11/24 7/24 4/24 2/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 8/24 5/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 2/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 11/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 0/24 */24 
agent 2: 11/24 */24 2/24 */24 
agent 3: */24 0/24 11/24 11/24 
agent 4: 0/24 8/24 11/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 0/24 */24 
agent 2: 11/24 */24 2/24 */24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 8/24 11/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 10/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: */24 5/24 */24 6/24 
agent 3: 2/24 5/24 11/24 6/24 
agent 4: 0/24 7/24 11/24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 10/24 */24 */24 4/24 
agent 3: */24 6/24 11/24 5/24 
agent 4: */24 0/24 11/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 10/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 4/24 
agent 2: 11/24 */24 */24 4/24 
agent 3: 2/24 10/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 5/24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 4/24 
agent 2: 11/24 */24 2/24 */24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 8/24 11/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 10/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: 5/24 8/24 */24 4/24 
agent 3: 5/24 8/24 0/24 11/24 
agent 4: 0/24 8/24 12/24 4/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 */24 4/24 
agent 3: 2/24 11/24 0/24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 */24 2/24 */24 
agent 3: */24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 5/24 6/24 
agent 2: 11/24 */24 5/24 6/24 
agent 3: 2/24 10/24 6/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 5/24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 4/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 6/24 */24 
agent 4: 2/24 11/24 0/24 11/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 5/24 
agent 3: 6/24 8/24 0/24 */24 
agent 4: */24 0/24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 5/24 
agent 3: 6/24 8/24 0/24 */24 
agent 4: */24 0/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 5/24 
agent 3: 6/24 8/24 0/24 10/24 
agent 4: */24 0/24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 8/24 */24 5/24 
agent 2: */24 8/24 */24 5/24 
agent 3: 6/24 8/24 0/24 10/24 
agent 4: */24 0/24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 5/24 
agent 2: 5/24 8/24 */24 4/24 
agent 3: 5/24 8/24 0/24 11/24 
agent 4: 0/24 8/24 12/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 5/24 
agent 2: 6/24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 */24 0/24 */24 
agent 3: */24 */24 */24 5/24 
agent 4: 6/24 */24 11/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: 6/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: */24 0/24 16/24 2/24 
agent 4: 6/24 0/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > b > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 4/24 
agent 4: 6/24 6/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 0/24 */24 5/24 
agent 2: 5/24 8/24 */24 4/24 
agent 3: 5/24 8/24 0/24 11/24 
agent 4: 0/24 8/24 12/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 5/24 5/24 
agent 2: 5/24 8/24 7/24 4/24 
agent 3: 5/24 8/24 0/24 11/24 
agent 4: 0/24 8/24 12/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 5/24 5/24 
agent 2: */24 8/24 */24 5/24 
agent 3: 6/24 8/24 0/24 10/24 
agent 4: */24 0/24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 5/24 
agent 2: 5/24 */24 8/24 */24 
agent 3: 5/24 0/24 8/24 11/24 
agent 4: 0/24 12/24 8/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 4/24 
agent 2: 11/24 */24 7/24 4/24 
agent 3: 2/24 10/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 8/24 */24 4/24 
agent 4: */24 8/24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 8/24 5/24 5/24 
agent 2: */24 8/24 */24 5/24 
agent 3: 6/24 8/24 0/24 10/24 
agent 4: */24 0/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 5/24 5/24 
agent 2: */24 8/24 */24 5/24 
agent 3: 6/24 8/24 0/24 10/24 
agent 4: */24 0/24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 5/24 
agent 2: 6/24 */24 */24 5/24 
agent 3: */24 */24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 0/24 5/24 
agent 2: 5/24 */24 8/24 */24 
agent 3: 5/24 0/24 8/24 11/24 
agent 4: 0/24 12/24 8/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 5/24 0/24 5/24 
agent 2: 5/24 */24 8/24 */24 
agent 3: 5/24 0/24 8/24 11/24 
agent 4: 0/24 12/24 8/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 5/24 0/24 5/24 
agent 2: 5/24 7/24 8/24 4/24 
agent 3: 5/24 0/24 8/24 11/24 
agent 4: 0/24 12/24 8/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 5/24 
agent 2: 6/24 4/24 8/24 6/24 
agent 3: */24 9/24 8/24 */24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 14/24 5/24 */24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: 5/24 0/24 11/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 4/24 
agent 2: 11/24 7/24 2/24 4/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 8/24 11/24 5/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 */24 8/24 4/24 
agent 4: */24 11/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 5/24 */24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: 5/24 0/24 11/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 5/24 5/24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: 5/24 0/24 11/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 5/24 5/24 0/24 
agent 2: 5/24 7/24 4/24 8/24 
agent 3: 5/24 0/24 11/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 5/24 0/24 
agent 2: 6/24 4/24 6/24 8/24 
agent 3: */24 9/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 7/24 4/24 2/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 8/24 5/24 11/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: */24 */24 4/24 8/24 
agent 4: */24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 7/24 4/24 2/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 8/24 5/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: 11/24 7/24 4/24 2/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 8/24 5/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 1/24 4/24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 7/24 2/24 4/24 
agent 3: 2/24 0/24 11/24 11/24 
agent 4: 0/24 8/24 11/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 4/24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: 0/24 6/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 1/24 4/24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 4/24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 4/24 0/24 
agent 2: 1/24 17/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 1/24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 5/24 0/24 
agent 2: 6/24 4/24 6/24 8/24 
agent 3: 1/24 9/24 6/24 8/24 
agent 4: 0/24 */24 */24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 17/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 */24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 1/24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 1/24 5/24 10/24 8/24 
agent 4: 0/24 6/24 10/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 5/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 1/24 5/24 8/24 10/24 
agent 4: 0/24 6/24 8/24 10/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 5/24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 5/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 5/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 17/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 6/24 8/24 6/24 4/24 
agent 3: 1/24 8/24 5/24 10/24 
agent 4: 0/24 8/24 6/24 10/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 5/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 5/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 5/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 */24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 5/24 */24 
agent 3: 6/24 */24 5/24 */24 
agent 4: */24 0/24 7/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 5/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 */24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 1/24 4/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 17/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 5/24 
agent 2: 6/24 */24 */24 5/24 
agent 3: */24 2/24 */24 */24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 0/24 1/24 4/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 17/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 1/24 4/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 1/24 0/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: */24 17/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 6/24 8/24 4/24 6/24 
agent 3: 1/24 8/24 9/24 6/24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: */24 17/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 5/24 
agent 2: 6/24 4/24 8/24 6/24 
agent 3: 1/24 9/24 8/24 6/24 
agent 4: 0/24 */24 8/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 17/24 0/24 6/24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 5/24 
agent 2: 6/24 4/24 8/24 6/24 
agent 3: 1/24 9/24 8/24 6/24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 */24 0/24 5/24 
agent 2: 6/24 4/24 8/24 6/24 
agent 3: 1/24 9/24 8/24 6/24 
agent 4: 0/24 */24 8/24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 2/24 0/24 5/24 
agent 2: 6/24 4/24 8/24 6/24 
agent 3: 1/24 9/24 8/24 6/24 
agent 4: 0/24 9/24 8/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 5/24 
agent 2: 6/24 8/24 4/24 6/24 
agent 3: 1/24 8/24 9/24 6/24 
agent 4: 0/24 8/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 2/24 5/24 0/24 
agent 2: 6/24 4/24 6/24 8/24 
agent 3: 1/24 9/24 6/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 5/24 */24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 */24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: */24 9/24 8/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 0/24 */24 5/24 
agent 2: 6/24 8/24 4/24 6/24 
agent 3: 1/24 8/24 9/24 6/24 
agent 4: 0/24 8/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 2/24 5/24 
agent 2: 6/24 8/24 4/24 6/24 
agent 3: 1/24 8/24 9/24 6/24 
agent 4: 0/24 8/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 2/24 5/24 
agent 2: 6/24 8/24 4/24 6/24 
agent 3: 1/24 8/24 9/24 6/24 
agent 4: 0/24 8/24 9/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 2/24 5/24 
agent 2: 6/24 */24 */24 5/24 
agent 3: */24 2/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 0/24 
agent 2: 6/24 8/24 6/24 4/24 
agent 3: 1/24 8/24 5/24 10/24 
agent 4: 0/24 8/24 6/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 6/24 
agent 3: 2/24 8/24 8/24 6/24 
agent 4: 0/24 8/24 8/24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 5/24 8/24 */24 6/24 
agent 4: 0/24 8/24 10/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 0/24 8/24 9/24 */24 
agent 4: 0/24 8/24 9/24 7/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 7/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: */24 8/24 */24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: */24 8/24 9/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 2/24 5/24 
agent 2: 6/24 */24 */24 5/24 
agent 3: */24 2/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 11/24 2/24 5/24 
agent 2: 6/24 */24 */24 5/24 
agent 3: */24 2/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 11/24 2/24 5/24 
agent 2: 6/24 11/24 */24 5/24 
agent 3: */24 2/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 11/24 2/24 5/24 
agent 2: 6/24 11/24 2/24 5/24 
agent 3: */24 2/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 11/24 */24 0/24 
agent 2: */24 */24 5/24 */24 
agent 3: 6/24 */24 5/24 */24 
agent 4: */24 0/24 7/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 1/24 */24 6/24 
agent 3: 8/24 0/24 */24 7/24 
agent 4: 0/24 */24 2/24 5/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 5/24 
agent 3: 6/24 5/24 8/24 5/24 
agent 4: 6/24 0/24 8/24 10/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 5/24 
agent 2: 6/24 11/24 0/24 */24 
agent 3: 6/24 2/24 11/24 */24 
agent 4: 6/24 0/24 11/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 11/24 */24 0/24 
agent 2: */24 */24 5/24 */24 
agent 3: 6/24 */24 5/24 */24 
agent 4: */24 0/24 7/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: */24 */24 5/24 */24 
agent 3: 6/24 */24 5/24 */24 
agent 4: */24 0/24 7/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: 6/24 8/24 6/24 4/24 
agent 3: 1/24 8/24 5/24 10/24 
agent 4: 0/24 8/24 6/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 5/24 */24 
agent 4: 0/24 */24 5/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 5/24 
agent 3: 6/24 */24 5/24 5/24 
agent 4: 6/24 0/24 4/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: */24 */24 12/24 */24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: 6/24 */24 */24 8/24 
agent 4: */24 */24 */24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 1/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 */24 2/24 12/24 
agent 4: 6/24 */24 */24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 */24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: 6/24 0/24 1/24 17/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: 6/24 */24 4/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 */24 12/24 4/24 
agent 3: 6/24 */24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 1/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 */24 2/24 12/24 
agent 4: 6/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 1/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 1/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 17/24 1/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 7/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 17/24 0/24 1/24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: 6/24 */24 12/24 */24 
agent 4: */24 1/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 7/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: 6/24 */24 4/24 12/24 
agent 4: */24 2/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 7/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 0/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 11/24 2/24 5/24 
agent 2: 6/24 11/24 0/24 */24 
agent 3: 6/24 2/24 11/24 */24 
agent 4: 6/24 0/24 11/24 7/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 */24 5/24 */24 
agent 3: 6/24 */24 5/24 */24 
agent 4: */24 0/24 7/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 1/24 
agent 2: 6/24 */24 9/24 1/24 
agent 3: 6/24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 6/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 16/24 */24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 11/24 2/24 5/24 
agent 2: 6/24 11/24 0/24 */24 
agent 3: 6/24 2/24 11/24 */24 
agent 4: 6/24 0/24 11/24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 11/24 2/24 5/24 
agent 2: 6/24 11/24 0/24 7/24 
agent 3: 6/24 2/24 11/24 5/24 
agent 4: 6/24 0/24 11/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 5/24 
agent 2: 6/24 8/24 5/24 5/24 
agent 3: */24 */24 0/24 10/24 
agent 4: */24 0/24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 1/24 */24 5/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 0/24 7/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 17/24 0/24 1/24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: 6/24 */24 12/24 */24 
agent 4: 6/24 1/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 1/24 */24 5/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 0/24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 1/24 10/24 5/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 17/24 0/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 10/24 5/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 17/24 0/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 4/24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 10/24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 18/24 0/24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 10/24 5/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 17/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 1/24 */24 6/24 
agent 3: 8/24 0/24 */24 7/24 
agent 4: 0/24 17/24 2/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 4/24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 0/24 
agent 2: 12/24 */24 10/24 0/24 
agent 3: 0/24 10/24 */24 12/24 
agent 4: 0/24 10/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 0/24 
agent 2: 12/24 */24 10/24 0/24 
agent 3: 0/24 10/24 */24 12/24 
agent 4: 0/24 10/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 2/24 10/24 0/24 
agent 3: 0/24 10/24 2/24 12/24 
agent 4: 0/24 10/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 */24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 10/24 4/24 10/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 */24 */24 1/24 
agent 3: 2/24 10/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 2/24 10/24 0/24 
agent 3: 0/24 10/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 10/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 17/24 7/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 */24 7/24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 0/24 6/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 17/24 */24 0/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 5/24 */24 
agent 4: 0/24 17/24 5/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 5/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 */24 5/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 */24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 */24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 */24 5/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 5/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 5/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 4/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 5/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 */24 7/24 0/24 
agent 3: 6/24 2/24 */24 12/24 
agent 4: 6/24 0/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 4/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 5/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 4/24 1/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 5/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 4/24 1/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 2/24 5/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 */24 5/24 */24 
agent 3: 6/24 2/24 5/24 */24 
agent 4: */24 0/24 7/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 5/24 17/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 */24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 1/24 4/24 */24 
agent 3: 8/24 0/24 5/24 */24 
agent 4: 0/24 17/24 5/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 */24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 4/24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 */24 1/24 
agent 4: 0/24 */24 */24 19/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 4/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 9/24 4/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 9/24 4/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 */24 
agent 2: */24 0/24 */24 11/24 
agent 3: 2/24 7/24 11/24 4/24 
agent 4: 0/24 9/24 11/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 */24 0/24 
agent 2: 11/24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 10/24 0/24 12/24 */24 
agent 3: */24 7/24 4/24 */24 
agent 4: */24 */24 4/24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 9/24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 0/24 */24 
agent 2: */24 0/24 */24 11/24 
agent 3: 2/24 7/24 11/24 4/24 
agent 4: 0/24 9/24 11/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 8/24 0/24 5/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 7/24 11/24 4/24 
agent 4: 0/24 9/24 11/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 5/24 
agent 2: 0/24 */24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 6/24 5/24 
agent 2: */24 */24 0/24 11/24 
agent 3: */24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 9/24 4/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 0/24 */24 
agent 2: 11/24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: 11/24 5/24 2/24 6/24 
agent 3: 2/24 5/24 11/24 6/24 
agent 4: 0/24 7/24 11/24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 11/24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 10/24 0/24 */24 12/24 
agent 3: */24 7/24 */24 4/24 
agent 4: */24 */24 11/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 5/24 
agent 2: 0/24 */24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 0/24 5/24 
agent 2: 0/24 17/24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 5/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 7/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 17/24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 17/24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 5/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 */24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 1/24 5/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 1/24 5/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 17/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 17/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 17/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 1/24 5/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 1/24 5/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 5/24 
agent 2: 0/24 17/24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 12/24 */24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 5/24 
agent 2: 0/24 17/24 0/24 */24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 0/24 5/24 
agent 2: 0/24 17/24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 8/24 0/24 5/24 
agent 2: 11/24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 0/24 17/24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 8/24 0/24 5/24 
agent 2: 11/24 0/24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 8/24 0/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 5/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: 11/24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 11/24 */24 */24 5/24 
agent 2: */24 0/24 0/24 */24 
agent 3: */24 */24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 5/24 
agent 2: */24 0/24 0/24 13/24 
agent 3: */24 */24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: */24 */24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 */24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 5/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 1/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 6/24 5/24 
agent 2: 11/24 */24 0/24 11/24 
agent 3: */24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 6/24 */24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: */24 11/24 8/24 */24 
agent 4: */24 11/24 8/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 5/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 6/24 5/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 6/24 5/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 6/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 */24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 3/24 
agent 3: */24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 6/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 */24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 */24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 1/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 11/24 1/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 1/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 */24 12/24 
agent 4: 0/24 11/24 1/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 */24 
agent 4: */24 11/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 1/24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 1/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 1/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 */24 0/24 
agent 2: */24 12/24 */24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 1/24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: */24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 3/24 
agent 3: 0/24 6/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 1/24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 1/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 11/24 8/24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 3/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 1/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 0/24 1/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 0/24 1/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 */24 0/24 
agent 2: 8/24 2/24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 */24 18/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 */24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 1/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 15/24 */24 0/24 
agent 2: 8/24 2/24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 */24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 6/24 */24 14/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 */24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 2/24 18/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 2/24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 3/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 */24 12/24 9/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 3/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 */24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 1/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 3/24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 */24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 */24 12/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 11/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 9/24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 */24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 23/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 */24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 */24 12/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 */24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 */24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 2/24 */24 
agent 3: 0/24 10/24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 9/24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 */24 12/24 9/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 0/24 3/24 */24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 */24 9/24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 */24 2/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 6/24 8/24 3/24 7/24 
agent 3: 0/24 8/24 */24 1/24 
agent 4: 0/24 8/24 2/24 14/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: */24 0/24 */24 1/24 
agent 4: 6/24 0/24 1/24 17/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 7/24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 */24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: 6/24 0/24 1/24 17/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 7/24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 3/24 */24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 3/24 7/24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 2/24 */24 
agent 3: 0/24 10/24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 10/24 0/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 3/24 7/24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 2/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 10/24 0/24 14/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 */24 0/24 10/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 10/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 2/24 */24 
agent 3: 0/24 10/24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 2/24 7/24 
agent 3: 0/24 10/24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 3/24 8/24 7/24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 */24 */24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 */24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 3/24 0/24 
agent 2: 0/24 23/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 3/24 */24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 11/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 3/24 0/24 
agent 2: 0/24 23/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 6/24 3/24 7/24 8/24 
agent 3: 0/24 */24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 23/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 6/24 3/24 8/24 7/24 
agent 3: 0/24 */24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 23/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 6/24 3/24 8/24 7/24 
agent 3: 0/24 */24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 6/24 3/24 8/24 7/24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: 6/24 3/24 8/24 7/24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 3/24 8/24 7/24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 2/24 
agent 2: 6/24 8/24 3/24 7/24 
agent 3: 0/24 8/24 */24 1/24 
agent 4: 0/24 8/24 2/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: 6/24 3/24 7/24 8/24 
agent 3: 0/24 */24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 15/24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: */24 3/24 8/24 7/24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 6/24 3/24 8/24 7/24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: 6/24 3/24 8/24 7/24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 2/24 
agent 2: */24 8/24 3/24 7/24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: */24 3/24 */24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 3/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 2/24 
agent 2: */24 8/24 3/24 7/24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: 6/24 8/24 3/24 7/24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 4/24 2/24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: 6/24 0/24 1/24 17/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 3/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 4/24 2/24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: 6/24 0/24 1/24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: 6/24 0/24 1/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 2/24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: */24 0/24 */24 1/24 
agent 4: 6/24 0/24 1/24 17/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: */24 12/24 */24 */24 
agent 3: 6/24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 */24 2/24 12/24 
agent 4: 6/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 */24 
agent 3: 6/24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 12/24 */24 2/24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: */24 0/24 */24 1/24 
agent 4: 6/24 0/24 1/24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: */24 0/24 */24 1/24 
agent 4: 6/24 0/24 1/24 17/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: 6/24 0/24 1/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: 6/24 8/24 3/24 7/24 
agent 3: 0/24 8/24 */24 1/24 
agent 4: 0/24 8/24 2/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: */24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 */24 5/24 */24 
agent 3: 6/24 2/24 5/24 */24 
agent 4: 6/24 0/24 7/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 17/24 */24 
agent 4: 6/24 0/24 */24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > d > a
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: 6/24 8/24 3/24 7/24 
agent 3: 0/24 8/24 */24 1/24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: 6/24 8/24 3/24 7/24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 3/24 */24 
agent 3: 0/24 8/24 15/24 */24 
agent 4: 0/24 8/24 2/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 3/24 */24 
agent 3: 0/24 8/24 15/24 */24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 3/24 */24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 0/24 4/24 */24 
agent 2: */24 8/24 3/24 */24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 4/24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 17/24 */24 
agent 4: 6/24 0/24 */24 17/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: 5/24 8/24 */24 1/24 
agent 3: 5/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 4/24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 17/24 */24 
agent 4: 6/24 0/24 */24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 12/24 4/24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: 6/24 0/24 1/24 17/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 17/24 1/24 
agent 4: 6/24 0/24 1/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 2/24 
agent 2: */24 8/24 3/24 */24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 2/24 
agent 2: 6/24 12/24 */24 */24 
agent 3: 6/24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: */24 0/24 16/24 2/24 
agent 4: 6/24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: 6/24 8/24 */24 1/24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 6/24 */24 1/24 */24 
agent 4: 6/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 2/24 
agent 2: */24 8/24 3/24 */24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: */24 8/24 3/24 */24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: 6/24 8/24 3/24 7/24 
agent 3: 0/24 8/24 15/24 1/24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 2/24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 15/24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 2/24 
agent 2: */24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 3/24 7/24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 */24 2/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 2/24 
agent 2: 6/24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 2/24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 15/24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: */24 */24 8/24 */24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: 6/24 3/24 8/24 7/24 
agent 3: 0/24 15/24 8/24 1/24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 */24 7/24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 11/24 11/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 3/24 8/24 12/24 
agent 4: 1/24 3/24 8/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 6/24 */24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 */24 
agent 2: 5/24 */24 8/24 1/24 
agent 3: 5/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 */24 0/24 2/24 
agent 2: 6/24 */24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: 6/24 7/24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 2/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 7/24 2/24 3/24 
agent 3: 0/24 8/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 12/24 8/24 3/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 2/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: 6/24 8/24 7/24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 */24 */24 
agent 3: 6/24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 2/24 7/24 3/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 10/24 0/24 14/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 12/24 3/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 */24 */24 
agent 3: 6/24 0/24 */24 2/24 
agent 4: */24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 */24 */24 
agent 3: 6/24 0/24 16/24 2/24 
agent 4: */24 0/24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 4/24 */24 
agent 3: 6/24 0/24 16/24 2/24 
agent 4: 6/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 16/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: 6/24 */24 12/24 2/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 16/24 2/24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 6/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 16/24 */24 
agent 4: 6/24 0/24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: 6/24 */24 12/24 2/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: */24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 */24 0/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 2/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 0/24 
agent 2: 6/24 */24 12/24 4/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: 6/24 2/24 4/24 12/24 
agent 4: */24 2/24 4/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 6/24 */24 8/24 */24 
agent 4: 6/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 0/24 1/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 2/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 0/24 20/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 3/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 1/24 8/24 12/24 3/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: */24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 1/24 11/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 1/24 12/24 8/24 3/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: */24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 1/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 1/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 1/24 11/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 3/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 3/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 3/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: */24 8/24 3/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 0/24 8/24 16/24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 1/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 */24 2/24 12/24 
agent 4: 6/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 */24 
agent 2: 1/24 8/24 12/24 3/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 1/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 */24 2/24 12/24 
agent 4: 6/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 17/24 1/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 17/24 0/24 */24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 17/24 0/24 */24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 17/24 0/24 1/24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 17/24 0/24 1/24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 1/24 0/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 1/24 8/24 12/24 3/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 1/24 
agent 2: 6/24 8/24 */24 1/24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 6/24 */24 8/24 */24 
agent 4: 6/24 1/24 8/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 17/24 0/24 1/24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: 6/24 */24 12/24 */24 
agent 4: 6/24 1/24 0/24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 8/24 */24 1/24 
agent 2: 6/24 8/24 */24 1/24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: */24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 9/24 1/24 
agent 2: 5/24 8/24 */24 1/24 
agent 3: 5/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 0/24 9/24 1/24 
agent 2: 5/24 8/24 */24 1/24 
agent 3: 5/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 9/24 1/24 
agent 2: 5/24 8/24 10/24 1/24 
agent 3: 5/24 8/24 */24 */24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 9/24 1/24 
agent 2: 5/24 8/24 10/24 1/24 
agent 3: 5/24 8/24 3/24 8/24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 1/24 
agent 2: 5/24 */24 8/24 1/24 
agent 3: 5/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 10/24 1/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 2/24 
agent 2: */24 */24 3/24 8/24 
agent 3: */24 10/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: */24 8/24 3/24 12/24 
agent 4: */24 8/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 0/24 1/24 
agent 2: 5/24 */24 8/24 1/24 
agent 3: 5/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 9/24 0/24 1/24 
agent 2: 5/24 10/24 8/24 1/24 
agent 3: 5/24 */24 8/24 */24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 9/24 0/24 1/24 
agent 2: 5/24 10/24 8/24 1/24 
agent 3: 5/24 3/24 8/24 8/24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 1/24 
agent 2: 1/24 12/24 8/24 3/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 9/24 */24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: 5/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 10/24 */24 1/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 1/24 11/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: */24 3/24 */24 8/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 11/24 11/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 3/24 8/24 12/24 
agent 4: */24 */24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 1/24 
agent 2: 1/24 12/24 8/24 3/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 1/24 12/24 8/24 3/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 */24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 9/24 1/24 0/24 
agent 2: 5/24 */24 */24 8/24 
agent 3: 5/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 7/24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 7/24 3/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 11/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: 6/24 3/24 */24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 2/24 0/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 2/24 0/24 
agent 2: 6/24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 2/24 0/24 
agent 2: 6/24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 7/24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: 6/24 */24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: 6/24 */24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 0/24 2/24 
agent 2: 6/24 7/24 8/24 3/24 
agent 3: 0/24 13/24 8/24 3/24 
agent 4: 0/24 0/24 8/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 7/24 */24 3/24 
agent 3: 0/24 8/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 7/24 */24 3/24 
agent 3: 0/24 8/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 7/24 2/24 3/24 
agent 3: 0/24 8/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 2/24 7/24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 11/24 11/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 8/24 */24 3/24 
agent 3: */24 6/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 2/24 7/24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 11/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 2/24 7/24 
agent 3: 0/24 10/24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 17/24 7/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: */24 3/24 */24 8/24 
agent 3: */24 */24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 17/24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 17/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 17/24 7/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 17/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 18/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 2/24 7/24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 10/24 0/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 7/24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 10/24 0/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 7/24 3/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 10/24 0/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 7/24 3/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 7/24 2/24 3/24 
agent 3: 0/24 8/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 */24 3/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 10/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 7/24 2/24 3/24 
agent 3: 0/24 8/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 7/24 2/24 3/24 
agent 3: 0/24 8/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 7/24 3/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 0/24 */24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 23/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 */24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: */24 0/24 */24 1/24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 1/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 3/24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 */24 */24 1/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 3/24 
agent 2: 8/24 0/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 11/24 8/24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 */24 3/24 
agent 3: 0/24 11/24 8/24 */24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 */24 0/24 3/24 
agent 3: 0/24 6/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 9/24 */24 
agent 3: 0/24 11/24 8/24 */24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 */24 0/24 3/24 
agent 3: 0/24 6/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 9/24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 9/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 */24 0/24 
agent 2: */24 12/24 1/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 23/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 12/24 */24 0/24 
agent 2: */24 12/24 1/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 12/24 1/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 */24 1/24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 */24 0/24 
agent 2: 11/24 12/24 */24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: */24 20/24 3/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 1/24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 1/24 20/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 1/24 20/24 3/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 12/24 1/24 0/24 
agent 2: 11/24 12/24 */24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 */24 0/24 0/24 
agent 2: 1/24 */24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 */24 
agent 2: 1/24 20/24 0/24 */24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 12/24 1/24 0/24 
agent 2: 11/24 12/24 */24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 12/24 1/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 */24 
agent 2: */24 12/24 0/24 */24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 */24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: */24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: 11/24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 8/24 5/24 0/24 
agent 2: 11/24 0/24 13/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 5/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 0/24 */24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: 11/24 0/24 13/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 5/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 5/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 5/24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 0/24 */24 
agent 2: 0/24 17/24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 1/24 0/24 */24 
agent 2: 0/24 17/24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 0/24 5/24 
agent 2: 0/24 17/24 0/24 7/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 0/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 5/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 17/24 7/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 17/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 8/24 0/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 8/24 0/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 0/24 0/24 13/24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: 2/24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 18/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: 2/24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 0/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 7/24 */24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 7/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: 11/24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 7/24 */24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 7/24 6/24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 5/24 6/24 
agent 2: 11/24 0/24 7/24 6/24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 5/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 6/24 
agent 2: 11/24 */24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: */24 12/24 */24 6/24 
agent 4: */24 0/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 5/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 5/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 5/24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 0/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 */24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: */24 12/24 */24 6/24 
agent 4: */24 0/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 */24 0/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 6/24 */24 
agent 2: 8/24 */24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 1/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 6/24 */24 
agent 2: 8/24 */24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 6/24 5/24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 5/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 6/24 5/24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 */24 5/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 0/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 1/24 5/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 17/24 7/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 16/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 1/24 5/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 */24 */24 
agent 4: 0/24 */24 17/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 */24 18/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 */24 5/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 3/24 
agent 4: 0/24 */24 */24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 3/24 
agent 4: 0/24 */24 */24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 */24 */24 3/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 3/24 
agent 4: 0/24 11/24 10/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 3/24 
agent 4: 0/24 11/24 10/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 4/24 6/24 
agent 2: 8/24 */24 10/24 5/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 11/24 1/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 4/24 6/24 
agent 2: 8/24 */24 10/24 5/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 10/24 5/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 10/24 5/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 17/24 0/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 10/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 18/24 0/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 */24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 17/24 */24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 10/24 5/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 17/24 0/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 */24 5/24 
agent 2: 8/24 */24 0/24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 */24 5/24 
agent 2: 8/24 */24 0/24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 0/24 17/24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 0/24 17/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 0/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 1/24 5/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 19/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 */24 5/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: */24 */24 0/24 3/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 6/24 */24 
agent 4: 0/24 0/24 17/24 7/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 */24 5/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 1/24 5/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 1/24 5/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 1/24 5/24 
agent 4: 0/24 0/24 17/24 7/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 */24 0/24 3/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 */24 0/24 3/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 10/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 10/24 0/24 3/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: 11/24 12/24 0/24 1/24 
agent 3: 2/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 10/24 0/24 3/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 10/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 12/24 0/24 1/24 
agent 3: 2/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 12/24 0/24 1/24 
agent 3: 2/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: 11/24 12/24 0/24 1/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: 11/24 12/24 0/24 1/24 
agent 3: 2/24 0/24 12/24 */24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: 11/24 12/24 0/24 1/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 17/24 7/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 11/24 1/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 17/24 7/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 5/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 5/24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 5/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 5/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 17/24 7/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 5/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 5/24 14/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 */24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 */24 5/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 5/24 5/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 7/24 4/24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 5/24 5/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 7/24 4/24 11/24 
agent 4: 0/24 9/24 */24 11/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 5/24 5/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 7/24 4/24 11/24 
agent 4: 0/24 9/24 */24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 8/24 5/24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 7/24 4/24 11/24 
agent 4: 0/24 9/24 4/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 5/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 */24 4/24 11/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 1/24 5/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 5/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 5/24 0/24 
agent 2: 0/24 19/24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 5/24 0/24 
agent 2: 0/24 19/24 5/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 2/24 */24 0/24 
agent 2: 6/24 0/24 10/24 8/24 
agent 3: 1/24 11/24 4/24 8/24 
agent 4: 0/24 */24 5/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 19/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 2/24 */24 0/24 
agent 2: 6/24 0/24 10/24 8/24 
agent 3: 1/24 11/24 4/24 8/24 
agent 4: 0/24 */24 5/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 2/24 5/24 0/24 
agent 2: 6/24 0/24 10/24 8/24 
agent 3: 1/24 11/24 4/24 8/24 
agent 4: 0/24 11/24 5/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 5/24 5/24 0/24 
agent 2: 5/24 7/24 4/24 8/24 
agent 3: */24 0/24 11/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: */24 */24 4/24 8/24 
agent 4: */24 11/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 5/24 5/24 0/24 
agent 2: 5/24 7/24 4/24 8/24 
agent 3: */24 0/24 11/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 5/24 5/24 0/24 
agent 2: 5/24 7/24 4/24 8/24 
agent 3: 5/24 0/24 11/24 8/24 
agent 4: 0/24 12/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: */24 */24 1/24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 5/24 */24 */24 8/24 
agent 4: 0/24 10/24 6/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 5/24 */24 */24 8/24 
agent 4: 0/24 10/24 6/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 5/24 */24 */24 8/24 
agent 4: 0/24 10/24 6/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 19/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 5/24 0/24 
agent 2: 0/24 19/24 5/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 5/24 0/24 
agent 2: 11/24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 */24 0/24 
agent 2: 0/24 19/24 5/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 8/24 5/24 0/24 
agent 2: 11/24 0/24 */24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 8/24 5/24 0/24 
agent 2: 11/24 0/24 13/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: 11/24 0/24 13/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: 11/24 0/24 13/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 8/24 5/24 0/24 
agent 2: 11/24 0/24 13/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 5/24 0/24 
agent 2: 0/24 19/24 5/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 5/24 0/24 
agent 2: 0/24 19/24 5/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 5/24 0/24 
agent 2: 0/24 19/24 5/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 0/24 */24 
agent 2: 0/24 19/24 0/24 5/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 1/24 0/24 */24 
agent 2: 0/24 19/24 0/24 5/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 0/24 5/24 
agent 2: 0/24 19/24 0/24 5/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: */24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: */24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 1/24 */24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: */24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 1/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 */24 6/24 
agent 4: */24 0/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 1/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 0/24 */24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 1/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 1/24 */24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 1/24 4/24 19/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 1/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 5/24 17/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 4/24 */24 
agent 4: 0/24 */24 */24 19/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 1/24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 1/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 5/24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 11/24 4/24 1/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 2/24 5/24 17/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 4/24 1/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 2/24 5/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 5/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 5/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 */24 */24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 */24 4/24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 5/24 5/24 14/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 */24 0/24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 4/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 2/24 */24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 5/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 */24 4/24 
agent 4: */24 0/24 5/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 5/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 19/24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 5/24 0/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 */24 */24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 17/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: */24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 17/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 */24 8/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 0/24 5/24 */24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 5/24 0/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 5/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: */24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 0/24 
agent 2: */24 5/24 0/24 */24 
agent 3: 2/24 3/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 */24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: */24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 1/24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 */24 */24 0/24 
agent 2: */24 17/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 */24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 1/24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 */24 */24 0/24 
agent 2: */24 17/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 */24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 */24 */24 0/24 
agent 2: 1/24 17/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 4/24 0/24 
agent 2: 1/24 17/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 11/24 */24 4/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 1/24 0/24 */24 
agent 2: */24 17/24 0/24 6/24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 1/24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 11/24 */24 4/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 11/24 9/24 4/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: 11/24 9/24 4/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 */24 10/24 0/24 
agent 3: 2/24 */24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 */24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 1/24 */24 0/24 
agent 2: */24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 9/24 */24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 */24 10/24 0/24 
agent 3: 2/24 */24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 3/24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 2/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 */24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 2/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: */24 4/24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 4/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 */24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 20/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: */24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: */24 */24 0/24 */24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 11/24 3/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 3/24 0/24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 */24 8/24 
agent 3: */24 */24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 */24 0/24 10/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 11/24 3/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 */24 2/24 
agent 2: */24 0/24 */24 10/24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 11/24 3/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: */24 0/24 */24 10/24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 9/24 2/24 
agent 2: 8/24 */24 */24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 */24 4/24 
agent 3: 2/24 11/24 0/24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 2/24 
agent 2: */24 0/24 */24 10/24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: 2/24 11/24 9/24 2/24 
agent 4: */24 11/24 3/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 9/24 2/24 
agent 2: 8/24 */24 */24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 */24 */24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 2/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 2/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 */24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 2/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 2/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 */24 */24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 */24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 7/24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 0/24 */24 4/24 
agent 3: 2/24 11/24 0/24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 */24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 0/24 */24 4/24 
agent 3: 2/24 11/24 0/24 11/24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 */24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 0/24 */24 4/24 
agent 3: 2/24 11/24 0/24 11/24 
agent 4: 0/24 11/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: 2/24 11/24 0/24 11/24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: 2/24 11/24 0/24 11/24 
agent 4: 0/24 11/24 8/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: 11/24 0/24 9/24 */24 
agent 3: 2/24 11/24 0/24 11/24 
agent 4: 0/24 */24 */24 5/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 4/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 6/24 5/24 
agent 4: 2/24 11/24 0/24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 11/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 9/24 0/24 */24 
agent 3: 2/24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 9/24 0/24 4/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: */24 17/24 0/24 6/24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 19/24 0/24 4/24 
agent 3: */24 0/24 12/24 */24 
agent 4: 2/24 4/24 12/24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 */24 
agent 2: 11/24 9/24 0/24 */24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 6/24 12/24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 */24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 10/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: 11/24 */24 0/24 6/24 
agent 3: 2/24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 10/24 0/24 4/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: */24 17/24 0/24 6/24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 */24 4/24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 */24 */24 
agent 4: */24 0/24 17/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 1/24 17/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 */24 12/24 8/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 */24 4/24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: */24 0/24 17/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 0/24 */24 4/24 
agent 2: 4/24 12/24 */24 6/24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: */24 0/24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 1/24 4/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 1/24 0/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 10/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: */24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 */24 6/24 
agent 3: */24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 1/24 0/24 17/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 10/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 2/24 16/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 2/24 16/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 2/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: 11/24 */24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: 11/24 */24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: 11/24 7/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 6/24 
agent 2: 11/24 7/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 0/24 6/24 
agent 2: 11/24 7/24 0/24 */24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: 11/24 7/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 17/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 */24 0/24 
agent 2: 11/24 7/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 17/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 17/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 1/24 0/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 */24 0/24 6/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 */24 6/24 
agent 4: 1/24 0/24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 */24 0/24 6/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 17/24 1/24 6/24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 2/24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 1/24 6/24 0/24 
agent 2: */24 */24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 6/24 0/24 
agent 2: 1/24 17/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 17/24 6/24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 1/24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 17/24 6/24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 1/24 17/24 6/24 0/24 
agent 3: */24 */24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: 1/24 */24 6/24 8/24 
agent 4: */24 */24 6/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 17/24 */24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: 1/24 */24 6/24 8/24 
agent 4: */24 */24 6/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: 1/24 9/24 6/24 8/24 
agent 4: */24 */24 6/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 2/24 5/24 0/24 
agent 2: 6/24 4/24 6/24 8/24 
agent 3: 1/24 9/24 6/24 8/24 
agent 4: 0/24 9/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 2/24 5/24 0/24 
agent 2: 6/24 4/24 6/24 8/24 
agent 3: 1/24 9/24 6/24 8/24 
agent 4: 0/24 9/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 2/24 5/24 0/24 
agent 2: 6/24 4/24 6/24 8/24 
agent 3: 1/24 9/24 6/24 8/24 
agent 4: 0/24 9/24 7/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 */24 0/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 1/24 5/24 10/24 8/24 
agent 4: 0/24 6/24 10/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 5/24 */24 6/24 8/24 
agent 4: 0/24 10/24 6/24 8/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: */24 4/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: 1/24 9/24 6/24 8/24 
agent 4: */24 9/24 6/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 */24 0/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 1/24 5/24 10/24 8/24 
agent 4: 0/24 6/24 10/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 1/24 5/24 10/24 8/24 
agent 4: 0/24 6/24 10/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 1/24 5/24 8/24 10/24 
agent 4: 0/24 6/24 8/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 5/24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: */24 5/24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: 5/24 5/24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 5/24 6/24 */24 
agent 3: 2/24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 */24 0/24 
agent 2: 5/24 */24 1/24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 5/24 6/24 */24 
agent 3: 2/24 3/24 8/24 */24 
agent 4: 0/24 3/24 10/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 5/24 6/24 2/24 
agent 3: 2/24 3/24 8/24 11/24 
agent 4: 0/24 3/24 10/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 6/24 2/24 
agent 3: 2/24 3/24 8/24 11/24 
agent 4: 0/24 3/24 10/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 */24 */24 2/24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 */24 2/24 6/24 
agent 3: 2/24 3/24 11/24 8/24 
agent 4: 0/24 3/24 11/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 6/24 8/24 
agent 3: 5/24 5/24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 5/24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 */24 1/24 2/24 
agent 3: 2/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 */24 11/24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 */24 */24 */24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 2/24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 */24 2/24 6/24 
agent 3: 2/24 3/24 11/24 8/24 
agent 4: 0/24 3/24 11/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 2/24 6/24 
agent 3: 2/24 3/24 11/24 8/24 
agent 4: 0/24 3/24 11/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 */24 */24 6/24 
agent 3: 2/24 10/24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 4/24 8/24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 0/24 
agent 2: 11/24 5/24 0/24 */24 
agent 3: 2/24 3/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 5/24 5/24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: 5/24 10/24 8/24 1/24 
agent 3: 5/24 3/24 8/24 8/24 
agent 4: 0/24 2/24 8/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 1/24 
agent 2: 5/24 10/24 8/24 1/24 
agent 3: 5/24 3/24 8/24 8/24 
agent 4: 0/24 2/24 8/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 9/24 0/24 1/24 
agent 2: 5/24 10/24 8/24 1/24 
agent 3: 5/24 3/24 8/24 8/24 
agent 4: 0/24 2/24 8/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 1/24 
agent 2: */24 8/24 */24 1/24 
agent 3: 5/24 8/24 3/24 8/24 
agent 4: 0/24 8/24 2/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 9/24 */24 0/24 
agent 2: 5/24 */24 1/24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 0/24 */24 1/24 
agent 2: */24 8/24 */24 1/24 
agent 3: 5/24 8/24 3/24 8/24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 9/24 1/24 
agent 2: */24 8/24 */24 1/24 
agent 3: 5/24 8/24 3/24 8/24 
agent 4: 0/24 8/24 2/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 9/24 1/24 
agent 2: 5/24 8/24 10/24 1/24 
agent 3: 5/24 8/24 3/24 8/24 
agent 4: 0/24 8/24 2/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 9/24 1/24 
agent 2: 6/24 */24 9/24 1/24 
agent 3: 6/24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 */24 10/24 1/24 
agent 3: 2/24 10/24 2/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 5/24 8/24 */24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 9/24 1/24 
agent 2: 6/24 */24 9/24 1/24 
agent 3: 6/24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: 6/24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: 6/24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: 6/24 8/24 */24 */24 
agent 4: */24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: 6/24 8/24 */24 */24 
agent 4: 6/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 5/24 
agent 2: 6/24 8/24 5/24 5/24 
agent 3: */24 */24 0/24 10/24 
agent 4: */24 0/24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 8/24 */24 5/24 
agent 3: 6/24 */24 5/24 5/24 
agent 4: 6/24 0/24 4/24 14/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: */24 8/24 8/24 */24 
agent 3: 6/24 8/24 8/24 2/24 
agent 4: */24 0/24 0/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 5/24 
agent 2: 6/24 8/24 5/24 5/24 
agent 3: */24 */24 0/24 10/24 
agent 4: 6/24 0/24 */24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: */24 8/24 */24 */24 
agent 4: 6/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 8/24 */24 5/24 
agent 2: 6/24 8/24 5/24 5/24 
agent 3: */24 */24 0/24 10/24 
agent 4: 6/24 0/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 5/24 5/24 
agent 2: 6/24 8/24 5/24 5/24 
agent 3: */24 */24 0/24 10/24 
agent 4: 6/24 0/24 14/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 5/24 5/24 
agent 2: 6/24 8/24 5/24 5/24 
agent 3: 6/24 8/24 0/24 10/24 
agent 4: 6/24 0/24 14/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 5/24 5/24 
agent 2: */24 8/24 */24 4/24 
agent 3: 5/24 8/24 0/24 11/24 
agent 4: 0/24 8/24 12/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 6/24 8/24 */24 */24 
agent 4: 6/24 0/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 5/24 5/24 
agent 2: */24 8/24 */24 5/24 
agent 3: 6/24 8/24 0/24 10/24 
agent 4: 6/24 0/24 14/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 0/24 5/24 5/24 
agent 2: */24 8/24 */24 4/24 
agent 3: 5/24 8/24 0/24 11/24 
agent 4: 0/24 8/24 12/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 5/24 5/24 
agent 2: 5/24 8/24 7/24 4/24 
agent 3: 5/24 8/24 0/24 11/24 
agent 4: 0/24 8/24 12/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 */24 7/24 4/24 
agent 3: */24 10/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: 5/24 8/24 2/24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 */24 7/24 4/24 
agent 3: */24 10/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 2/24 7/24 4/24 
agent 3: 2/24 10/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 2/24 7/24 4/24 
agent 3: 2/24 10/24 0/24 12/24 
agent 4: 0/24 10/24 10/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 0/24 */24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 11/24 8/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 2/24 2/24 */24 
agent 3: 2/24 10/24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 2/24 */24 */24 
agent 3: 2/24 10/24 6/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 */24 10/24 1/24 
agent 3: 2/24 10/24 2/24 */24 
agent 4: 0/24 10/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 4/24 
agent 2: 11/24 */24 7/24 4/24 
agent 3: 2/24 10/24 0/24 12/24 
agent 4: 0/24 10/24 10/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 0/24 */24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 11/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 11/24 8/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: 2/24 11/24 0/24 11/24 
agent 4: 0/24 11/24 8/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 11/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 */24 6/24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 11/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: 11/24 0/24 9/24 */24 
agent 3: 2/24 11/24 0/24 11/24 
agent 4: 0/24 11/24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: 11/24 0/24 9/24 */24 
agent 3: 2/24 11/24 0/24 11/24 
agent 4: 0/24 11/24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 */24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: 2/24 11/24 0/24 11/24 
agent 4: 0/24 11/24 8/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: 2/24 11/24 0/24 11/24 
agent 4: 0/24 11/24 8/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 7/24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 4/24 
agent 2: 11/24 9/24 0/24 */24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 6/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 2/24 
agent 2: */24 0/24 */24 10/24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 4/24 
agent 2: */24 0/24 */24 4/24 
agent 3: */24 11/24 6/24 5/24 
agent 4: 2/24 11/24 0/24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: */24 11/24 0/24 */24 
agent 4: 0/24 11/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 7/24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 7/24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 2/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 5/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 2/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 2/24 16/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 */24 4/24 
agent 2: 8/24 11/24 */24 4/24 
agent 3: 8/24 0/24 */24 10/24 
agent 4: 0/24 2/24 16/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 2/24 16/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 1/24 4/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 1/24 0/24 17/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 2/24 */24 17/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 2/24 17/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 9/24 0/24 4/24 
agent 3: */24 0/24 12/24 10/24 
agent 4: 0/24 6/24 12/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 16/24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 1/24 4/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 */24 8/24 
agent 4: 1/24 0/24 17/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 1/24 4/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 1/24 0/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 */24 0/24 4/24 
agent 2: 1/24 17/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 */24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 1/24 0/24 17/24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: 11/24 0/24 */24 6/24 
agent 3: */24 11/24 */24 6/24 
agent 4: 0/24 11/24 7/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 17/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 */24 0/24 4/24 
agent 2: 1/24 17/24 0/24 */24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 */24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 0/24 4/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 1/24 */24 0/24 
agent 2: 1/24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: */24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 1/24 */24 0/24 
agent 2: 1/24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 1/24 4/24 0/24 
agent 2: 1/24 17/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 4/24 0/24 
agent 2: 11/24 9/24 */24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 */24 0/24 0/24 
agent 2: 1/24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 */24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 4/24 0/24 
agent 2: 11/24 9/24 */24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: 11/24 9/24 4/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: 11/24 9/24 4/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 9/24 0/24 */24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 6/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 1/24 */24 0/24 
agent 2: */24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 10/24 4/24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 11/24 9/24 4/24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: 0/24 6/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 9/24 0/24 */24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 9/24 0/24 4/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: */24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 17/24 1/24 */24 0/24 
agent 2: */24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 1/24 0/24 */24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 2/24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 0/24 */24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 1/24 0/24 19/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 1/24 6/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: */24 12/24 4/24 6/24 
agent 4: 1/24 0/24 19/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 1/24 6/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 2/24 17/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 0/24 7/24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 2/24 17/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 9/24 0/24 7/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 17/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 */24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 7/24 
agent 4: 0/24 */24 18/24 5/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 9/24 */24 6/24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 16/24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 */24 
agent 3: 8/24 0/24 6/24 */24 
agent 4: 0/24 2/24 17/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 17/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 5/24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 5/24 6/24 
agent 2: 11/24 0/24 */24 6/24 
agent 3: */24 11/24 */24 6/24 
agent 4: 0/24 11/24 7/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 5/24 6/24 
agent 2: 11/24 0/24 */24 6/24 
agent 3: */24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 4/24 
agent 2: 8/24 5/24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 2/24 5/24 6/24 
agent 2: 11/24 0/24 */24 6/24 
agent 3: */24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 5/24 6/24 
agent 2: 11/24 0/24 7/24 6/24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 0/24 
agent 2: 11/24 0/24 */24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 7/24 6/24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: 2/24 11/24 0/24 */24 
agent 4: 0/24 11/24 8/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: 11/24 7/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 */24 0/24 
agent 2: 11/24 */24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 7/24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 0/24 6/24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 11/24 9/24 0/24 4/24 
agent 3: 2/24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 7/24 */24 0/24 
agent 2: 11/24 */24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 17/24 1/24 6/24 0/24 
agent 2: */24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 0/24 
agent 2: 11/24 */24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 11/24 9/24 4/24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: 0/24 6/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 1/24 6/24 0/24 
agent 2: */24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 6/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 1/24 */24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 */24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: */24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 */24 0/24 8/24 
agent 3: 2/24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 */24 0/24 8/24 
agent 3: 2/24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 5/24 0/24 8/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 0/24 0/24 
agent 2: 11/24 5/24 0/24 8/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 */24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 0/24 8/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 */24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: 1/24 5/24 0/24 18/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 */24 
agent 2: 11/24 12/24 0/24 */24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 3/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 */24 0/24 */24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 */24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 1/24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 1/24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 17/24 */24 */24 0/24 
agent 2: 1/24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: */24 6/24 12/24 4/24 
agent 4: 1/24 5/24 0/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 1/24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 */24 */24 0/24 
agent 2: 1/24 17/24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 */24 */24 0/24 
agent 2: 1/24 17/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 6/24 0/24 
agent 2: 1/24 17/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 */24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: */24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: */24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 1/24 0/24 17/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 1/24 0/24 19/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 1/24 0/24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 1/24 0/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 1/24 6/24 
agent 2: 8/24 9/24 */24 6/24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 16/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 1/24 6/24 
agent 2: 8/24 9/24 */24 6/24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 16/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 9/24 1/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 16/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 */24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 */24 17/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 1/24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 7/24 0/24 6/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 16/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 9/24 */24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 */24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 5/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 5/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: */24 0/24 5/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: */24 0/24 5/24 */24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: */24 0/24 5/24 */24 
agent 3: */24 11/24 4/24 */24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 */24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: */24 0/24 5/24 */24 
agent 3: */24 11/24 4/24 7/24 
agent 4: 0/24 11/24 4/24 9/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: */24 0/24 5/24 */24 
agent 3: */24 11/24 4/24 7/24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: */24 0/24 5/24 */24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 0/24 5/24 8/24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: 11/24 0/24 */24 */24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 11/24 1/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 0/24 5/24 8/24 
agent 3: */24 11/24 4/24 */24 
agent 4: 0/24 11/24 4/24 9/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 2/24 
agent 2: */24 0/24 */24 10/24 
agent 3: 2/24 11/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: 11/24 0/24 */24 */24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 11/24 1/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 */24 */24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 11/24 1/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 0/24 */24 1/24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 1/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 0/24 */24 1/24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 1/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 0/24 12/24 1/24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 1/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 0/24 12/24 1/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 11/24 1/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 */24 10/24 1/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 12/24 */24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 11/24 1/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 11/24 1/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 0/24 */24 3/24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 */24 10/24 1/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 2/24 5/24 6/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 2/24 5/24 6/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 2/24 5/24 6/24 
agent 3: 2/24 10/24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 10/24 4/24 */24 
agent 4: 0/24 10/24 4/24 10/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 2/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 2/24 10/24 */24 2/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 10/24 4/24 */24 
agent 4: 0/24 10/24 4/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 10/24 4/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 10/24 4/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 10/24 4/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 10/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 1/24 4/24 */24 
agent 4: 0/24 */24 6/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 6/24 4/24 
agent 3: */24 8/24 5/24 */24 
agent 4: */24 8/24 */24 10/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 10/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 10/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 1/24 4/24 */24 
agent 4: 0/24 */24 6/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 */24 6/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 1/24 4/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 5/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 */24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 1/24 4/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 5/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 1/24 4/24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 5/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 4/24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 5/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 2/24 */24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 10/24 4/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 1/24 */24 6/24 
agent 3: 8/24 0/24 */24 7/24 
agent 4: 0/24 17/24 2/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 11/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 */24 5/24 2/24 
agent 3: 6/24 2/24 5/24 */24 
agent 4: 6/24 0/24 7/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 5/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > d > a > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 2/24 */24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 10/24 4/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 10/24 4/24 10/24 
agent 4: 0/24 10/24 4/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 */24 6/24 
agent 3: 8/24 0/24 */24 7/24 
agent 4: 0/24 17/24 2/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 */24 6/24 
agent 3: 8/24 0/24 */24 7/24 
agent 4: 0/24 17/24 2/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 9/24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 17/24 2/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 9/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 9/24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 17/24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 9/24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 9/24 7/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 17/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 9/24 7/24 
agent 3: 8/24 0/24 */24 7/24 
agent 4: 0/24 18/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 17/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 9/24 7/24 
agent 3: 8/24 0/24 */24 7/24 
agent 4: 0/24 18/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 9/24 7/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 18/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 17/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 17/24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 16/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 */24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 3/24 
agent 3: 8/24 1/24 5/24 */24 
agent 4: 0/24 1/24 14/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 10/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 1/24 5/24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 16/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 */24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 3/24 
agent 2: 8/24 5/24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 8/24 3/24 
agent 2: 8/24 5/24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 */24 1/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 8/24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 */24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 6/24 5/24 
agent 2: 8/24 5/24 6/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 8/24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 11/24 3/24 */24 
agent 4: 0/24 11/24 11/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 0/24 2/24 11/24 
agent 3: 2/24 11/24 3/24 8/24 
agent 4: 0/24 11/24 11/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: */24 0/24 */24 3/24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 */24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 1/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: */24 0/24 */24 3/24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 */24 3/24 
agent 2: 11/24 0/24 */24 3/24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 0/24 10/24 3/24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 10/24 0/24 3/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 */24 8/24 3/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 0/24 10/24 3/24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 6/24 5/24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 */24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 18/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 1/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 */24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 1/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 1/24 3/24 12/24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 1/24 3/24 12/24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 */24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 1/24 23/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 9/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 1/24 23/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 */24 1/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 20/24 4/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 */24 1/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 11/24 1/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 11/24 1/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 10/24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 1/24 3/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 10/24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 1/24 3/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 1/24 3/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 1/24 3/24 12/24 
agent 4: 0/24 16/24 8/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 */24 0/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 11/24 
agent 4: 0/24 16/24 6/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 1/24 3/24 12/24 
agent 4: 0/24 16/24 8/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 */24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 */24 0/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 10/24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 */24 0/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 10/24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 9/24 6/24 
agent 4: 0/24 16/24 */24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 10/24 */24 
agent 4: 0/24 16/24 0/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 9/24 6/24 
agent 4: 0/24 16/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 */24 */24 
agent 4: 0/24 1/24 17/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 */24 */24 
agent 4: 0/24 1/24 17/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 */24 6/24 
agent 4: 0/24 1/24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 */24 */24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 2/24 12/24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 */24 2/24 */24 
agent 3: 2/24 5/24 11/24 6/24 
agent 4: 0/24 7/24 11/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 */24 17/24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 4/24 */24 
agent 4: 0/24 */24 */24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 */24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 4/24 */24 
agent 4: 0/24 */24 */24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 */24 19/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 */24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 1/24 */24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 */24 */24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 17/24 6/24 */24 
agent 4: 0/24 1/24 */24 19/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 4/24 11/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 19/24 */24 1/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 */24 4/24 9/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 4/24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 1/24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 */24 */24 19/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 */24 6/24 */24 
agent 4: 0/24 1/24 4/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 11/24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 */24 11/24 
agent 4: 0/24 16/24 6/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 0/24 5/24 */24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: 0/24 11/24 4/24 9/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 11/24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 0/24 5/24 */24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 0/24 5/24 8/24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 */24 5/24 6/24 
agent 3: 2/24 10/24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 0/24 
agent 2: 11/24 0/24 */24 8/24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 9/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 */24 
agent 3: 0/24 11/24 4/24 */24 
agent 4: 0/24 11/24 4/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 */24 5/24 6/24 
agent 3: 2/24 10/24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 2/24 5/24 6/24 
agent 3: 2/24 10/24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 2/24 5/24 6/24 
agent 3: 2/24 10/24 4/24 8/24 
agent 4: 0/24 10/24 4/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 11/24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 2/24 */24 */24 
agent 3: 2/24 10/24 6/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 5/24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: 2/24 10/24 2/24 */24 
agent 4: 0/24 10/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 11/24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 2/24 */24 */24 
agent 3: 2/24 10/24 6/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 5/24 6/24 
agent 2: 11/24 2/24 5/24 6/24 
agent 3: 2/24 10/24 6/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 5/24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 6/24 
agent 2: 11/24 */24 2/24 */24 
agent 3: 2/24 5/24 11/24 6/24 
agent 4: 0/24 7/24 11/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 5/24 6/24 
agent 3: 5/24 8/24 */24 6/24 
agent 4: 0/24 8/24 10/24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 10/24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 5/24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 5/24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 6/24 8/24 4/24 6/24 
agent 4: 6/24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > c > d > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 6/24 8/24 4/24 6/24 
agent 4: 6/24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 6/24 8/24 4/24 6/24 
agent 4: 6/24 0/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 4/24 6/24 
agent 2: 6/24 8/24 4/24 6/24 
agent 3: 6/24 8/24 4/24 6/24 
agent 4: 6/24 0/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 4/24 6/24 
agent 2: */24 8/24 5/24 6/24 
agent 3: 5/24 8/24 */24 6/24 
agent 4: 0/24 8/24 10/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 6/24 
agent 2: 6/24 */24 0/24 */24 
agent 3: */24 */24 */24 5/24 
agent 4: 6/24 */24 11/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 0/24 
agent 2: 6/24 8/24 */24 5/24 
agent 3: 6/24 */24 5/24 5/24 
agent 4: 6/24 0/24 4/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 11/24 2/24 5/24 
agent 2: 6/24 11/24 2/24 5/24 
agent 3: 6/24 2/24 */24 6/24 
agent 4: */24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 */24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: */24 8/24 8/24 */24 
agent 3: 6/24 8/24 8/24 2/24 
agent 4: 6/24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 4/24 6/24 
agent 2: */24 8/24 5/24 6/24 
agent 3: 5/24 8/24 */24 6/24 
agent 4: 0/24 8/24 10/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 4/24 6/24 
agent 2: 5/24 8/24 5/24 6/24 
agent 3: 5/24 8/24 5/24 6/24 
agent 4: 0/24 8/24 10/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 5/24 */24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 0/24 
agent 2: 5/24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 5/24 */24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 4/24 0/24 6/24 
agent 2: */24 */24 */24 */24 
agent 3: 5/24 5/24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 4/24 0/24 6/24 
agent 2: 5/24 5/24 8/24 6/24 
agent 3: 5/24 5/24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 0/24 8/24 */24 
agent 3: */24 */24 8/24 4/24 
agent 4: */24 11/24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 4/24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 5/24 */24 6/24 8/24 
agent 4: 0/24 10/24 6/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 6/24 
agent 2: 11/24 5/24 2/24 6/24 
agent 3: 2/24 5/24 11/24 6/24 
agent 4: 0/24 7/24 11/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 4/24 */24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 5/24 */24 6/24 8/24 
agent 4: 0/24 10/24 6/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 4/24 6/24 0/24 
agent 2: */24 */24 */24 8/24 
agent 3: 5/24 5/24 6/24 8/24 
agent 4: 0/24 10/24 6/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 4/24 6/24 0/24 
agent 2: 5/24 5/24 6/24 8/24 
agent 3: 5/24 5/24 6/24 8/24 
agent 4: 0/24 10/24 6/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: */24 */24 4/24 8/24 
agent 4: */24 11/24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 0/24 
agent 2: */24 5/24 6/24 8/24 
agent 3: 5/24 5/24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 5/24 6/24 2/24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 0/24 0/24 
agent 2: */24 5/24 6/24 8/24 
agent 3: 5/24 5/24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 10/24 0/24 0/24 
agent 2: 5/24 5/24 6/24 8/24 
agent 3: 5/24 5/24 6/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 10/24 0/24 0/24 
agent 2: 5/24 5/24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 10/24 0/24 0/24 
agent 2: 5/24 5/24 8/24 6/24 
agent 3: */24 */24 8/24 6/24 
agent 4: 0/24 4/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 10/24 0/24 0/24 
agent 2: 5/24 5/24 8/24 6/24 
agent 3: 5/24 5/24 8/24 6/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 0/24 
agent 2: 11/24 5/24 6/24 2/24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 5/24 6/24 2/24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: 11/24 5/24 2/24 6/24 
agent 3: 2/24 5/24 11/24 6/24 
agent 4: 0/24 7/24 11/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: */24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 19/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 1/24 6/24 0/24 
agent 2: */24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 6/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 1/24 0/24 */24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 1/24 */24 4/24 8/24 
agent 4: */24 11/24 */24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 1/24 17/24 6/24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 */24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: */24 */24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: */24 */24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 3/24 8/24 12/24 
agent 4: 1/24 3/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 7/24 0/24 
agent 2: 8/24 9/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 19/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: */24 8/24 3/24 12/24 
agent 4: 1/24 8/24 3/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 7/24 0/24 
agent 2: 8/24 9/24 */24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 */24 6/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 */24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 9/24 */24 6/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 17/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 17/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 19/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 2/24 */24 12/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 2/24 17/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 19/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 1/24 0/24 19/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 8/24 */24 4/24 
agent 4: 1/24 8/24 11/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: */24 19/24 0/24 */24 
agent 3: */24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: */24 19/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 1/24 */24 8/24 4/24 
agent 4: */24 11/24 8/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 17/24 0/24 6/24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 1/24 */24 8/24 4/24 
agent 4: */24 11/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: */24 11/24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 2/24 0/24 6/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: */24 11/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: */24 0/24 8/24 */24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: */24 11/24 8/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 8/24 */24 4/24 
agent 4: 1/24 8/24 11/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: */24 0/24 */24 8/24 
agent 3: 1/24 */24 4/24 8/24 
agent 4: */24 11/24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 8/24 */24 4/24 
agent 4: 1/24 8/24 11/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: */24 8/24 0/24 */24 
agent 3: */24 8/24 */24 4/24 
agent 4: 1/24 8/24 11/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: */24 8/24 0/24 10/24 
agent 3: */24 8/24 11/24 4/24 
agent 4: 1/24 8/24 11/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 6/24 8/24 0/24 10/24 
agent 3: 1/24 8/24 11/24 4/24 
agent 4: 1/24 8/24 11/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 2/24 6/24 
agent 2: 6/24 */24 0/24 */24 
agent 3: */24 */24 */24 5/24 
agent 4: 6/24 */24 11/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: */24 8/24 3/24 12/24 
agent 4: 1/24 8/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 10/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: 6/24 8/24 */24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: */24 8/24 9/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 2/24 6/24 
agent 2: 6/24 */24 0/24 */24 
agent 3: */24 */24 */24 5/24 
agent 4: 6/24 */24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 10/24 2/24 6/24 
agent 2: 6/24 */24 0/24 */24 
agent 3: */24 */24 */24 5/24 
agent 4: 6/24 */24 11/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 10/24 2/24 6/24 
agent 2: 6/24 */24 0/24 */24 
agent 3: 6/24 */24 11/24 5/24 
agent 4: 6/24 */24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 10/24 2/24 6/24 
agent 2: 6/24 */24 0/24 */24 
agent 3: 6/24 2/24 11/24 5/24 
agent 4: 6/24 */24 11/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 10/24 */24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 17/24 5/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 5/24 5/24 
agent 2: 6/24 8/24 */24 5/24 
agent 3: 6/24 8/24 0/24 10/24 
agent 4: 6/24 0/24 14/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 10/24 */24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 10/24 8/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 10/24 8/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: */24 8/24 3/24 12/24 
agent 4: 1/24 8/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: 6/24 2/24 4/24 12/24 
agent 4: */24 2/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: 6/24 8/24 */24 */24 
agent 4: 6/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: */24 8/24 3/24 12/24 
agent 4: 1/24 8/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: */24 8/24 */24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 1/24 8/24 3/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 6/24 8/24 10/24 0/24 
agent 3: 1/24 8/24 3/24 12/24 
agent 4: 1/24 8/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 3/24 8/24 12/24 
agent 4: 1/24 3/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 10/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 10/24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 6/24 8/24 6/24 4/24 
agent 3: */24 8/24 5/24 */24 
agent 4: */24 8/24 */24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: */24 3/24 8/24 12/24 
agent 4: 1/24 3/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 1/24 3/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 6/24 10/24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 1/24 3/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 0/24 8/24 
agent 3: */24 */24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 10/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 1/24 11/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 6/24 8/24 */24 
agent 3: */24 5/24 8/24 */24 
agent 4: */24 */24 8/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 10/24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 1/24 11/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 2/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 1/24 11/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 2/24 0/24 
agent 3: 0/24 1/24 11/24 12/24 
agent 4: 0/24 1/24 11/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 */24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 11/24 10/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 10/24 */24 1/24 
agent 3: */24 1/24 11/24 */24 
agent 4: 0/24 1/24 11/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 */24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 11/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 11/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 11/24 10/24 
agent 4: 0/24 3/24 11/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: */24 5/24 8/24 */24 
agent 4: */24 */24 8/24 10/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 */24 6/24 
agent 3: */24 3/24 11/24 8/24 
agent 4: 0/24 3/24 11/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: */24 5/24 8/24 */24 
agent 4: */24 */24 8/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: */24 5/24 8/24 10/24 
agent 4: */24 */24 8/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 1/24 5/24 8/24 10/24 
agent 4: */24 */24 8/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 5/24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: c > d > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 */24 8/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 5/24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 5/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 5/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: */24 6/24 */24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 1/24 5/24 8/24 10/24 
agent 4: 1/24 5/24 8/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 0/24 */24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 5/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 5/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 7/24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 5/24 17/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 17/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 6/24 8/24 6/24 4/24 
agent 3: */24 8/24 5/24 */24 
agent 4: 1/24 8/24 5/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 7/24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 5/24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 6/24 1/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 5/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 6/24 */24 
agent 3: 6/24 */24 5/24 */24 
agent 4: */24 2/24 5/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 9/24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 17/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: */24 0/24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 1/24 0/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 */24 0/24 6/24 
agent 2: */24 17/24 0/24 6/24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 6/24 
agent 2: 6/24 8/24 */24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 1/24 8/24 9/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 */24 0/24 6/24 
agent 2: */24 17/24 0/24 6/24 
agent 3: 4/24 */24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: 1/24 17/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 17/24 1/24 6/24 0/24 
agent 2: 1/24 17/24 6/24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 6/24 
agent 2: */24 */24 8/24 6/24 
agent 3: 1/24 */24 8/24 6/24 
agent 4: */24 9/24 8/24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 1/24 6/24 0/24 
agent 2: 1/24 17/24 6/24 0/24 
agent 3: 4/24 */24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 6/24 0/24 
agent 2: 1/24 17/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 */24 
agent 4: 0/24 1/24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 */24 
agent 4: 0/24 1/24 */24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 */24 */24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 */24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 4/24 9/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 */24 4/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 */24 4/24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 */24 17/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 */24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 */24 6/24 
agent 4: 0/24 */24 17/24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 1/24 17/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 17/24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 */24 17/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 1/24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 17/24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 1/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 */24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 1/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 16/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 */24 
agent 4: 0/24 */24 18/24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 */24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 11/24 7/24 0/24 6/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 */24 6/24 
agent 4: 0/24 1/24 17/24 6/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 */24 18/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 */24 6/24 
agent 4: 0/24 1/24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 */24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 */24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 1/24 18/24 5/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 */24 18/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 */24 18/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 */24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 0/24 7/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 17/24 5/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 */24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 */24 0/24 7/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 17/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 9/24 0/24 7/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 17/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 9/24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 10/24 2/24 6/24 
agent 2: 6/24 */24 0/24 */24 
agent 3: 6/24 2/24 11/24 5/24 
agent 4: 6/24 2/24 11/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 9/24 */24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 9/24 1/24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 */24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 */24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: a > c > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 10/24 2/24 6/24 
agent 2: 6/24 */24 0/24 */24 
agent 3: 6/24 2/24 11/24 5/24 
agent 4: 6/24 2/24 11/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 10/24 2/24 6/24 
agent 2: 6/24 10/24 0/24 8/24 
agent 3: 6/24 2/24 11/24 5/24 
agent 4: 6/24 2/24 11/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 */24 0/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 6/24 
agent 2: 6/24 10/24 */24 6/24 
agent 3: 6/24 */24 */24 6/24 
agent 4: */24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 */24 0/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 */24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 */24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 6/24 */24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 6/24 5/24 
agent 2: 8/24 5/24 6/24 5/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 0/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 */24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 10/24 1/24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: 2/24 11/24 9/24 2/24 
agent 4: */24 11/24 3/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 8/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 14/24 8/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: 2/24 11/24 9/24 2/24 
agent 4: */24 11/24 3/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: */24 0/24 2/24 */24 
agent 3: 2/24 11/24 9/24 2/24 
agent 4: 2/24 11/24 3/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 0/24 2/24 12/24 
agent 3: 2/24 11/24 9/24 2/24 
agent 4: 2/24 11/24 3/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 10/24 2/24 
agent 2: 8/24 4/24 */24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 */24 4/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 6/24 5/24 
agent 4: 2/24 11/24 0/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 10/24 */24 2/24 */24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: */24 */24 2/24 10/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: 10/24 0/24 2/24 12/24 
agent 3: */24 11/24 */24 2/24 
agent 4: 2/24 11/24 3/24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: */24 11/24 8/24 */24 
agent 4: 2/24 11/24 8/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 10/24 2/24 
agent 2: 8/24 4/24 */24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 2/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 */24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 */24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 8/24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 6/24 5/24 
agent 4: 2/24 11/24 0/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 11/24 6/24 5/24 
agent 4: 2/24 11/24 0/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: */24 0/24 */24 */24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 2/24 11/24 0/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 0/24 10/24 4/24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 2/24 11/24 0/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 4/24 
agent 2: 10/24 */24 */24 4/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 2/24 10/24 0/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 4/24 
agent 2: 10/24 0/24 10/24 4/24 
agent 3: */24 11/24 6/24 5/24 
agent 4: 2/24 11/24 0/24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: 2/24 11/24 1/24 */24 
agent 4: */24 11/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 */24 4/24 
agent 2: 10/24 */24 */24 4/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 2/24 10/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: 10/24 */24 8/24 4/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 2/24 10/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: 10/24 2/24 8/24 4/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 2/24 10/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 8/24 4/24 
agent 2: 8/24 4/24 8/24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 8/24 4/24 
agent 3: */24 8/24 */24 4/24 
agent 4: 4/24 8/24 0/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: 10/24 2/24 2/24 */24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: */24 */24 2/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 8/24 4/24 
agent 2: 8/24 4/24 8/24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 8/24 4/24 
agent 2: 8/24 4/24 8/24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 8/24 4/24 
agent 2: 8/24 4/24 8/24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 12/24 8/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > b > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 4/24 
agent 4: 6/24 6/24 0/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 8/24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 12/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 */24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 10/24 2/24 
agent 2: 10/24 2/24 2/24 */24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: */24 */24 2/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > d > c
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 10/24 2/24 
agent 2: 10/24 2/24 2/24 10/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: */24 */24 2/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 8/24 */24 */24 
agent 3: */24 8/24 2/24 10/24 
agent 4: */24 8/24 2/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: */24 */24 4/24 8/24 
agent 3: 2/24 10/24 4/24 8/24 
agent 4: 2/24 10/24 4/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 2/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 */24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 8/24 */24 */24 
agent 3: */24 8/24 2/24 10/24 
agent 4: */24 8/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 8/24 */24 */24 
agent 3: 4/24 8/24 2/24 10/24 
agent 4: */24 8/24 2/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 8/24 4/24 
agent 3: 4/24 8/24 */24 4/24 
agent 4: 4/24 8/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: */24 8/24 8/24 4/24 
agent 3: 4/24 8/24 */24 4/24 
agent 4: 4/24 8/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 0/24 */24 */24 
agent 2: 4/24 8/24 8/24 4/24 
agent 3: 4/24 8/24 8/24 4/24 
agent 4: 4/24 8/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 0/24 8/24 4/24 
agent 2: 4/24 8/24 8/24 4/24 
agent 3: 4/24 8/24 8/24 4/24 
agent 4: 4/24 8/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > b > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 4/24 
agent 4: 6/24 6/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 4/24 
agent 2: */24 */24 8/24 4/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 0/24 */24 */24 
agent 2: 4/24 8/24 */24 */24 
agent 3: 4/24 8/24 2/24 10/24 
agent 4: */24 8/24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > b > d > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: */24 */24 */24 */24 
agent 3: */24 */24 8/24 4/24 
agent 4: 6/24 6/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 6/24 8/24 4/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 6/24 6/24 8/24 4/24 
agent 4: 6/24 6/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 4/24 
agent 2: 6/24 */24 8/24 5/24 
agent 3: 6/24 5/24 8/24 5/24 
agent 4: 6/24 0/24 8/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > b > d > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 6/24 */24 */24 
agent 2: 6/24 6/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > b > d > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 6/24 */24 */24 
agent 2: 6/24 6/24 */24 */24 
agent 3: 6/24 6/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: */24 8/24 8/24 */24 
agent 3: */24 8/24 8/24 2/24 
agent 4: 6/24 0/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: a > c > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 0/24 4/24 
agent 2: 6/24 */24 8/24 5/24 
agent 3: 6/24 5/24 8/24 5/24 
agent 4: 6/24 0/24 8/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 14/24 0/24 4/24 
agent 2: 6/24 5/24 8/24 5/24 
agent 3: 6/24 5/24 8/24 5/24 
agent 4: 6/24 0/24 8/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 8/24 8/24 6/24 
agent 3: 2/24 8/24 8/24 6/24 
agent 4: 0/24 8/24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 14/24 4/24 0/24 
agent 2: */24 */24 5/24 8/24 
agent 3: */24 5/24 5/24 8/24 
agent 4: 6/24 0/24 10/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 8/24 8/24 6/24 
agent 3: 2/24 8/24 8/24 6/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: 2/24 8/24 8/24 6/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: */24 8/24 8/24 */24 
agent 4: 0/24 8/24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > a > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: */24 8/24 8/24 */24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: 2/24 8/24 8/24 6/24 
agent 4: 0/24 8/24 8/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 14/24 4/24 0/24 
agent 2: */24 */24 5/24 8/24 
agent 3: */24 5/24 5/24 8/24 
agent 4: 6/24 0/24 10/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 14/24 4/24 0/24 
agent 2: */24 */24 5/24 8/24 
agent 3: 6/24 5/24 5/24 8/24 
agent 4: 6/24 0/24 10/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 14/24 4/24 0/24 
agent 2: 6/24 5/24 5/24 8/24 
agent 3: 6/24 5/24 5/24 8/24 
agent 4: 6/24 0/24 10/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: 6/24 */24 */24 8/24 
agent 4: */24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: 6/24 */24 */24 8/24 
agent 4: */24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: 6/24 */24 */24 8/24 
agent 4: 6/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 6/24 */24 8/24 */24 
agent 4: 6/24 1/24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > c > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 6/24 */24 8/24 */24 
agent 4: 6/24 1/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 */24 
agent 3: 6/24 */24 8/24 */24 
agent 4: 6/24 1/24 8/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 1/24 5/24 10/24 
agent 4: 0/24 1/24 14/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 1/24 5/24 10/24 
agent 4: 0/24 1/24 14/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 8/24 5/24 3/24 
agent 3: 8/24 1/24 5/24 10/24 
agent 4: 0/24 1/24 14/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 10/24 2/24 */24 */24 
agent 3: 2/24 8/24 11/24 3/24 
agent 4: */24 2/24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 10/24 2/24 */24 */24 
agent 3: 2/24 8/24 11/24 3/24 
agent 4: */24 2/24 */24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 12/24 0/24 2/24 
agent 2: 10/24 2/24 */24 10/24 
agent 3: 2/24 8/24 11/24 3/24 
agent 4: */24 2/24 */24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 2/24 2/24 10/24 
agent 3: 2/24 8/24 11/24 3/24 
agent 4: */24 2/24 */24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 2/24 2/24 10/24 
agent 3: 2/24 8/24 11/24 3/24 
agent 4: 2/24 2/24 11/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 10/24 2/24 
agent 2: 10/24 2/24 2/24 10/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: */24 */24 2/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 2/24 */24 */24 
agent 3: 2/24 */24 3/24 11/24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 4/24 */24 8/24 */24 
agent 3: */24 2/24 8/24 10/24 
agent 4: */24 2/24 8/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 10/24 */24 2/24 4/24 
agent 3: */24 6/24 11/24 5/24 
agent 4: */24 0/24 11/24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 */24 8/24 
agent 3: 2/24 3/24 11/24 8/24 
agent 4: 2/24 3/24 11/24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 10/24 0/24 */24 12/24 
agent 3: */24 7/24 */24 4/24 
agent 4: 2/24 */24 11/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 10/24 2/24 
agent 2: 10/24 2/24 2/24 10/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: */24 */24 2/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 2/24 2/24 10/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: */24 */24 2/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 2/24 2/24 10/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 2/24 10/24 2/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 2/24 8/24 4/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 2/24 10/24 0/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 2/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 10/24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 8/24 4/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 2/24 10/24 8/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 2/24 8/24 4/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 2/24 10/24 0/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 2/24 8/24 4/24 
agent 3: 2/24 10/24 8/24 4/24 
agent 4: 2/24 10/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 0/24 4/24 
agent 2: 10/24 */24 2/24 4/24 
agent 3: */24 6/24 11/24 5/24 
agent 4: */24 0/24 11/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 */24 */24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 0/24 4/24 
agent 2: 10/24 */24 2/24 4/24 
agent 3: */24 6/24 11/24 5/24 
agent 4: */24 0/24 11/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 8/24 2/24 4/24 
agent 3: 2/24 6/24 11/24 5/24 
agent 4: 2/24 0/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 */24 4/24 */24 
agent 3: */24 6/24 5/24 11/24 
agent 4: */24 0/24 11/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 4/24 
agent 2: */24 8/24 8/24 4/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 4/24 0/24 8/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: */24 6/24 */24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: 2/24 5/24 11/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 */24 4/24 */24 
agent 3: */24 6/24 5/24 11/24 
agent 4: */24 0/24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 */24 4/24 */24 
agent 3: 2/24 6/24 5/24 11/24 
agent 4: */24 0/24 11/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 8/24 4/24 */24 
agent 3: 2/24 6/24 5/24 11/24 
agent 4: 2/24 0/24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 8/24 4/24 2/24 
agent 3: 2/24 6/24 5/24 11/24 
agent 4: 2/24 0/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 8/24 4/24 8/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 2/24 */24 2/24 
agent 3: 2/24 */24 3/24 11/24 
agent 4: */24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 */24 */24 11/24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: */24 6/24 6/24 */24 
agent 3: */24 5/24 6/24 */24 
agent 4: 2/24 5/24 6/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: */24 8/24 4/24 8/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 4/24 8/24 4/24 8/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 4/24 */24 */24 8/24 
agent 3: 4/24 */24 */24 8/24 
agent 4: */24 2/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 2/24 */24 2/24 
agent 3: 2/24 */24 3/24 11/24 
agent 4: */24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 2/24 10/24 2/24 
agent 3: 2/24 8/24 3/24 11/24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 2/24 10/24 2/24 
agent 3: 2/24 8/24 3/24 11/24 
agent 4: 2/24 2/24 9/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 4/24 */24 */24 8/24 
agent 3: 4/24 2/24 10/24 8/24 
agent 4: */24 2/24 */24 8/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 8/24 */24 
agent 3: 2/24 3/24 8/24 11/24 
agent 4: 2/24 3/24 8/24 11/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 */24 */24 14/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 */24 */24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 2/24 */24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 10/24 0/24 12/24 */24 
agent 3: */24 7/24 4/24 */24 
agent 4: 2/24 */24 4/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 8/24 */24 
agent 3: 2/24 3/24 8/24 11/24 
agent 4: 2/24 3/24 8/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 8/24 2/24 
agent 3: 2/24 3/24 8/24 11/24 
agent 4: 2/24 3/24 8/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: */24 3/24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 10/24 */24 */24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: */24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 */24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: */24 3/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 10/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: */24 3/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 10/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 2/24 3/24 7/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 10/24 4/24 10/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: */24 3/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 10/24 4/24 10/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: */24 3/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 14/24 0/24 0/24 
agent 2: 10/24 4/24 10/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: */24 3/24 7/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 10/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 2/24 3/24 7/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 0/24 
agent 2: 10/24 */24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: */24 */24 12/24 */24 
agent 4: */24 3/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 12/24 */24 0/24 
agent 3: */24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 */24 0/24 
agent 2: 10/24 */24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 */24 */24 0/24 
agent 2: 10/24 8/24 6/24 0/24 
agent 3: */24 */24 */24 12/24 
agent 4: */24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 */24 */24 0/24 
agent 2: 10/24 8/24 6/24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 2/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 2/24 */24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: 10/24 8/24 6/24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 2/24 */24 6/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 10/24 4/24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 2/24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 2/24 */24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: */24 */24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: 10/24 8/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 6/24 6/24 
agent 2: 10/24 0/24 8/24 6/24 
agent 3: */24 */24 5/24 6/24 
agent 4: 2/24 11/24 5/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 */24 0/24 
agent 2: 10/24 8/24 6/24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 2/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: 10/24 8/24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: */24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 6/24 6/24 
agent 2: 10/24 0/24 8/24 6/24 
agent 3: */24 */24 5/24 6/24 
agent 4: 2/24 11/24 5/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 6/24 6/24 
agent 2: 10/24 0/24 8/24 6/24 
agent 3: */24 */24 5/24 6/24 
agent 4: 2/24 11/24 5/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 6/24 6/24 
agent 2: 10/24 0/24 8/24 6/24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 2/24 11/24 5/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 4/24 7/24 
agent 4: */24 11/24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 */24 4/24 7/24 
agent 4: */24 11/24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: */24 11/24 4/24 7/24 
agent 4: */24 11/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: */24 0/24 4/24 */24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: */24 11/24 4/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: 10/24 0/24 2/24 12/24 
agent 3: 2/24 11/24 */24 2/24 
agent 4: 2/24 11/24 3/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: 10/24 0/24 2/24 12/24 
agent 3: 2/24 11/24 */24 2/24 
agent 4: 2/24 11/24 3/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 */24 */24 2/24 
agent 2: 10/24 0/24 2/24 12/24 
agent 3: 2/24 11/24 9/24 2/24 
agent 4: 2/24 11/24 3/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 0/24 2/24 12/24 
agent 3: 2/24 11/24 9/24 2/24 
agent 4: 2/24 11/24 3/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 */24 4/24 
agent 2: 10/24 0/24 10/24 4/24 
agent 3: */24 11/24 6/24 5/24 
agent 4: 2/24 11/24 0/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 2/24 
agent 2: */24 0/24 */24 10/24 
agent 3: 2/24 11/24 */24 2/24 
agent 4: 0/24 11/24 3/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 8/24 5/24 3/24 
agent 3: 8/24 1/24 5/24 10/24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 12/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 8/24 5/24 3/24 
agent 3: 8/24 1/24 5/24 10/24 
agent 4: 0/24 */24 14/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 8/24 5/24 3/24 
agent 3: 8/24 1/24 5/24 10/24 
agent 4: 0/24 1/24 14/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 */24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 3/24 
agent 3: 8/24 1/24 5/24 */24 
agent 4: 0/24 1/24 14/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 */24 
agent 4: 0/24 1/24 0/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 3/24 
agent 3: 8/24 1/24 5/24 */24 
agent 4: 0/24 1/24 14/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 8/24 5/24 3/24 
agent 3: 8/24 1/24 5/24 10/24 
agent 4: 0/24 1/24 14/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 */24 8/24 
agent 3: */24 8/24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 10/24 */24 1/24 
agent 3: */24 1/24 11/24 10/24 
agent 4: 0/24 1/24 11/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 */24 8/24 
agent 3: */24 8/24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 2/24 8/24 
agent 3: 2/24 8/24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 8/24 2/24 3/24 
agent 3: */24 6/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 */24 6/24 
agent 3: 2/24 3/24 11/24 8/24 
agent 4: 0/24 3/24 11/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 8/24 2/24 3/24 
agent 3: */24 6/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 8/24 2/24 3/24 
agent 3: 2/24 6/24 11/24 5/24 
agent 4: 0/24 0/24 11/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 10/24 */24 1/24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: 0/24 1/24 11/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: */24 10/24 */24 1/24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: 0/24 1/24 11/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: 11/24 10/24 2/24 1/24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: 0/24 1/24 11/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 10/24 1/24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: 11/24 12/24 0/24 1/24 
agent 3: */24 0/24 12/24 10/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 2/24 6/24 
agent 3: 2/24 3/24 11/24 8/24 
agent 4: 0/24 3/24 11/24 10/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: */24 1/24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 10/24 1/24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: */24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: 2/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 */24 0/24 11/24 
agent 3: */24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 */24 8/24 3/24 
agent 3: 2/24 */24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 */24 0/24 11/24 
agent 3: */24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: */24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 5/24 6/24 
agent 2: 11/24 2/24 5/24 6/24 
agent 3: 2/24 10/24 6/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 */24 2/24 */24 
agent 3: 2/24 */24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 */24 0/24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 */24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 11/24 4/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 11/24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 9/24 11/24 4/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 9/24 11/24 4/24 
agent 4: 0/24 9/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 2/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 */24 4/24 11/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 2/24 10/24 
agent 3: 0/24 */24 11/24 4/24 
agent 4: 0/24 9/24 11/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 2/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 18/24 0/24 6/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 4/24 2/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > c > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 4/24 12/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 1/24 5/24 
agent 3: 8/24 4/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 1/24 5/24 
agent 3: 8/24 4/24 */24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 1/24 5/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 1/24 5/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 6/24 */24 
agent 4: 0/24 0/24 16/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 11/24 2/24 5/24 
agent 2: 6/24 11/24 2/24 5/24 
agent 3: 6/24 2/24 */24 6/24 
agent 4: */24 0/24 */24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 9/24 1/24 6/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 16/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 6/24 */24 
agent 4: 0/24 0/24 16/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 10/24 
agent 4: */24 0/24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 */24 */24 
agent 3: */24 0/24 */24 */24 
agent 4: 6/24 0/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: */24 12/24 0/24 */24 
agent 3: */24 0/24 12/24 10/24 
agent 4: */24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: */24 12/24 0/24 */24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: */24 0/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 10/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 10/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 10/24 0/24 4/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 2/24 0/24 12/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 10/24 
agent 4: 0/24 2/24 16/24 6/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: 10/24 8/24 0/24 6/24 
agent 3: */24 */24 12/24 6/24 
agent 4: 2/24 */24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 */24 10/24 
agent 4: 0/24 2/24 16/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 1/24 4/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 2/24 16/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 9/24 1/24 6/24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 16/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 9/24 1/24 6/24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 16/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 9/24 1/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 16/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: 10/24 8/24 0/24 6/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 2/24 */24 12/24 6/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 6/24 
agent 2: 6/24 10/24 */24 6/24 
agent 3: 6/24 */24 */24 6/24 
agent 4: */24 2/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: 10/24 8/24 0/24 6/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 2/24 */24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: 10/24 8/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: 10/24 6/24 */24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: 2/24 5/24 11/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: 10/24 6/24 */24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: 2/24 5/24 11/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: 10/24 6/24 2/24 6/24 
agent 3: */24 5/24 */24 6/24 
agent 4: 2/24 5/24 11/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 8/24 0/24 6/24 
agent 2: 10/24 6/24 2/24 6/24 
agent 3: 2/24 5/24 11/24 6/24 
agent 4: 2/24 5/24 11/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 10/24 0/24 2/24 12/24 
agent 3: */24 7/24 */24 4/24 
agent 4: 2/24 */24 11/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 10/24 0/24 2/24 12/24 
agent 3: */24 7/24 */24 4/24 
agent 4: 2/24 */24 11/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 10/24 0/24 2/24 12/24 
agent 3: */24 7/24 */24 4/24 
agent 4: 2/24 7/24 11/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 0/24 4/24 
agent 2: 10/24 0/24 2/24 12/24 
agent 3: */24 7/24 11/24 4/24 
agent 4: 2/24 7/24 11/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 0/24 2/24 12/24 
agent 3: 2/24 7/24 11/24 4/24 
agent 4: 2/24 7/24 11/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 8/24 4/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 2/24 10/24 8/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 */24 0/24 
agent 2: 10/24 0/24 12/24 */24 
agent 3: */24 7/24 4/24 */24 
agent 4: 2/24 */24 4/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 8/24 4/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 2/24 10/24 8/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 2/24 10/24 8/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: 2/24 10/24 8/24 4/24 
agent 4: 2/24 10/24 8/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 */24 */24 
agent 4: */24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 */24 */24 4/24 
agent 4: 0/24 10/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 */24 */24 4/24 
agent 4: 0/24 10/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 */24 */24 4/24 
agent 4: 0/24 10/24 10/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 10/24 8/24 4/24 
agent 4: 0/24 10/24 10/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 8/24 4/24 
agent 2: 8/24 */24 8/24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 12/24 */24 4/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 10/24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 */24 2/24 */24 
agent 3: 2/24 10/24 2/24 */24 
agent 4: 0/24 10/24 12/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 */24 */24 4/24 
agent 4: 0/24 10/24 10/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 8/24 4/24 
agent 2: 8/24 */24 8/24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 12/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 8/24 4/24 
agent 2: 8/24 4/24 8/24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 12/24 8/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 */24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 */24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 4/24 
agent 2: 8/24 */24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 12/24 8/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 */24 */24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 */24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 */24 */24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 */24 */24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 */24 2/24 */24 
agent 3: 2/24 10/24 2/24 10/24 
agent 4: 0/24 10/24 12/24 2/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 */24 2/24 */24 
agent 3: 2/24 10/24 2/24 10/24 
agent 4: 0/24 10/24 12/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 2/24 2/24 9/24 
agent 3: 2/24 10/24 2/24 10/24 
agent 4: 0/24 10/24 12/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: 5/24 8/24 2/24 9/24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 */24 4/24 
agent 2: 11/24 2/24 7/24 4/24 
agent 3: 2/24 10/24 0/24 12/24 
agent 4: 0/24 10/24 10/24 4/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 3/24 7/24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: 5/24 8/24 2/24 9/24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: 5/24 8/24 2/24 9/24 
agent 3: 5/24 8/24 2/24 */24 
agent 4: 0/24 8/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: 5/24 8/24 2/24 9/24 
agent 3: 5/24 8/24 2/24 9/24 
agent 4: 0/24 8/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 */24 4/24 
agent 2: 11/24 2/24 7/24 4/24 
agent 3: 2/24 10/24 0/24 12/24 
agent 4: 0/24 10/24 10/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 2/24 7/24 4/24 
agent 3: 2/24 10/24 0/24 12/24 
agent 4: 0/24 10/24 10/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 7/24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 2/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 10/24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 7/24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 7/24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 5/24 5/24 
agent 2: 6/24 8/24 5/24 5/24 
agent 3: 6/24 8/24 0/24 10/24 
agent 4: 6/24 0/24 14/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 */24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 10/24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 */24 */24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 10/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: */24 */24 3/24 8/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 2/24 3/24 8/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 2/24 8/24 3/24 
agent 3: 2/24 */24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 2/24 3/24 8/24 
agent 3: 2/24 10/24 10/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: */24 11/24 */24 2/24 
agent 4: 0/24 11/24 3/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: 2/24 11/24 9/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 2/24 
agent 2: 11/24 0/24 */24 10/24 
agent 3: 2/24 11/24 */24 2/24 
agent 4: 0/24 11/24 3/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 */24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 0/24 5/24 8/24 
agent 3: 2/24 11/24 4/24 */24 
agent 4: 0/24 11/24 4/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 */24 2/24 
agent 2: 11/24 0/24 */24 10/24 
agent 3: 2/24 11/24 */24 2/24 
agent 4: 0/24 11/24 3/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: 2/24 11/24 9/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 3/24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 2/24 */24 */24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: */24 0/24 3/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 0/24 7/24 0/24 
agent 2: 4/24 12/24 8/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 1/24 0/24 19/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 17/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 9/24 */24 */24 
agent 2: 8/24 */24 0/24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 17/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 */24 
agent 2: 8/24 9/24 0/24 */24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 17/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 9/24 0/24 7/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 17/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 0/24 1/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 1/24 0/24 19/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 */24 18/24 5/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 1/24 6/24 
agent 2: 8/24 9/24 1/24 6/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 16/24 6/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 */24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 */24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 5/24 6/24 
agent 2: 8/24 5/24 5/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 14/24 4/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 5/24 6/24 
agent 2: 11/24 0/24 7/24 6/24 
agent 3: 2/24 11/24 5/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 */24 18/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 0/24 7/24 
agent 4: 0/24 1/24 18/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 */24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > a > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 */24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 */24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 4/24 8/24 12/24 0/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: */24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 17/24 1/24 */24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: */24 */24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 2/24 6/24 12/24 4/24 
agent 4: 1/24 5/24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 1/24 */24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 6/24 0/24 
agent 2: 1/24 19/24 4/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 1/24 0/24 */24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 2/24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 */24 0/24 0/24 
agent 2: 11/24 5/24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 0/24 */24 
agent 2: 11/24 7/24 0/24 6/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 */24 0/24 0/24 
agent 2: 11/24 5/24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 0/24 */24 
agent 3: 2/24 3/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 0/24 */24 
agent 3: 2/24 3/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 0/24 8/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 */24 0/24 8/24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 9/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 18/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 */24 0/24 8/24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 */24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 11/24 5/24 0/24 8/24 
agent 3: */24 3/24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 13/24 0/24 0/24 
agent 2: 11/24 5/24 0/24 8/24 
agent 3: */24 3/24 12/24 7/24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 0/24 8/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 */24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 5/24 0/24 18/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 18/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 */24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 8/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 19/24 */24 0/24 0/24 
agent 2: 1/24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 */24 0/24 0/24 
agent 2: 1/24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 1/24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 19/24 5/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 5/24 0/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 18/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 6/24 */24 
agent 3: 8/24 */24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 */24 18/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 1/24 23/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 1/24 */24 14/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 */24 12/24 
agent 4: 0/24 */24 23/24 0/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 23/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 */24 1/24 2/24 
agent 3: 2/24 */24 */24 */24 
agent 4: 0/24 1/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 */24 1/24 2/24 
agent 3: 2/24 */24 */24 */24 
agent 4: 0/24 1/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 2/24 
agent 3: 2/24 */24 */24 */24 
agent 4: 0/24 1/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 2/24 
agent 3: 2/24 1/24 */24 */24 
agent 4: 0/24 1/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 9/24 */24 0/24 
agent 2: 5/24 10/24 1/24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 */24 5/24 
agent 4: 0/24 1/24 */24 14/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 0/24 2/24 
agent 3: 0/24 1/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 9/24 */24 0/24 
agent 2: 5/24 10/24 1/24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 9/24 1/24 0/24 
agent 2: 5/24 10/24 1/24 8/24 
agent 3: */24 3/24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 9/24 1/24 0/24 
agent 2: 5/24 10/24 1/24 8/24 
agent 3: 5/24 3/24 8/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 1/24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 8/24 */24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 2/24 */24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 8/24 */24 0/24 
agent 2: */24 */24 3/24 */24 
agent 3: 5/24 */24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 1/24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: 6/24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: 6/24 */24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 7/24 3/24 8/24 
agent 3: 0/24 13/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 3/24 */24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 */24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 3/24 */24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 3/24 7/24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 2/24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 2/24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 2/24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 2/24 
agent 3: 0/24 8/24 5/24 11/24 
agent 4: 0/24 0/24 13/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: 2/24 6/24 5/24 11/24 
agent 4: 0/24 0/24 13/24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 7/24 3/24 */24 
agent 3: 0/24 8/24 */24 11/24 
agent 4: 0/24 0/24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 */24 3/24 */24 
agent 3: 2/24 6/24 5/24 11/24 
agent 4: 0/24 0/24 13/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 8/24 3/24 2/24 
agent 3: 2/24 6/24 5/24 11/24 
agent 4: 0/24 0/24 13/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 8/24 */24 0/24 
agent 2: */24 8/24 3/24 */24 
agent 3: 5/24 */24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 8/24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 2/24 */24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 2/24 
agent 3: 2/24 1/24 */24 11/24 
agent 4: 0/24 1/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 8/24 */24 0/24 
agent 2: */24 8/24 3/24 */24 
agent 3: 5/24 */24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 8/24 2/24 0/24 
agent 2: */24 8/24 3/24 */24 
agent 3: 5/24 8/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 8/24 2/24 0/24 
agent 2: 5/24 8/24 3/24 8/24 
agent 3: 5/24 8/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 4/24 0/24 
agent 2: 5/24 2/24 9/24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: */24 12/24 */24 8/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 1/24 12/24 3/24 8/24 
agent 3: 1/24 12/24 3/24 8/24 
agent 4: 0/24 0/24 16/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: 1/24 3/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 3/24 8/24 12/24 
agent 4: 1/24 3/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 3/24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > d > a > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 3/24 8/24 12/24 
agent 4: 1/24 3/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 1/24 3/24 8/24 12/24 
agent 4: 1/24 3/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 3/24 */24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 3/24 7/24 8/24 
agent 3: 0/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 3/24 7/24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 11/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 9/24 1/24 0/24 
agent 2: 5/24 */24 1/24 8/24 
agent 3: 5/24 */24 */24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: 6/24 3/24 7/24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 2/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 2/24 11/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 2/24 
agent 3: 0/24 10/24 3/24 11/24 
agent 4: 0/24 2/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 2/24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 */24 */24 
agent 3: */24 */24 3/24 11/24 
agent 4: 0/24 2/24 */24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 */24 
agent 3: 0/24 10/24 */24 11/24 
agent 4: 0/24 2/24 11/24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 2/24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 2/24 11/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 2/24 
agent 3: 0/24 */24 5/24 */24 
agent 4: 0/24 0/24 13/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 2/24 
agent 3: 0/24 8/24 5/24 11/24 
agent 4: 0/24 0/24 13/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 8/24 3/24 */24 
agent 3: */24 6/24 5/24 11/24 
agent 4: 0/24 0/24 13/24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 */24 
agent 3: 0/24 8/24 5/24 11/24 
agent 4: 0/24 0/24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 8/24 3/24 */24 
agent 3: */24 6/24 5/24 11/24 
agent 4: 0/24 0/24 13/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 8/24 3/24 2/24 
agent 3: 2/24 6/24 5/24 11/24 
agent 4: 0/24 0/24 13/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 */24 2/24 
agent 3: */24 */24 3/24 11/24 
agent 4: 0/24 2/24 */24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 */24 1/24 */24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 */24 2/24 
agent 3: */24 */24 3/24 11/24 
agent 4: 0/24 2/24 */24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 8/24 2/24 
agent 3: */24 */24 3/24 11/24 
agent 4: 0/24 2/24 11/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 8/24 2/24 
agent 3: 2/24 8/24 3/24 11/24 
agent 4: 0/24 2/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 14/24 9/24 1/24 0/24 
agent 2: 5/24 */24 1/24 8/24 
agent 3: 5/24 3/24 8/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 6/24 */24 
agent 3: 2/24 3/24 8/24 11/24 
agent 4: 0/24 3/24 10/24 11/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 8/24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 2/24 11/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 9/24 1/24 0/24 
agent 2: 5/24 */24 1/24 8/24 
agent 3: 5/24 3/24 8/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 9/24 1/24 0/24 
agent 2: 5/24 10/24 1/24 8/24 
agent 3: 5/24 3/24 8/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 */24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 0/24 8/24 
agent 3: */24 3/24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 */24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 13/24 0/24 0/24 
agent 2: 11/24 5/24 6/24 2/24 
agent 3: 2/24 3/24 8/24 11/24 
agent 4: 0/24 3/24 10/24 11/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 11/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 10/24 */24 */24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: */24 */24 */24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 2/24 
agent 3: 2/24 1/24 */24 11/24 
agent 4: 0/24 1/24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 11/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 1/24 12/24 11/24 
agent 4: 0/24 */24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 1/24 */24 0/24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 2/24 
agent 3: 2/24 1/24 */24 11/24 
agent 4: 0/24 */24 */24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 0/24 2/24 
agent 3: 0/24 1/24 12/24 */24 
agent 4: 0/24 */24 12/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 1/24 */24 0/24 
agent 4: 0/24 3/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: */24 3/24 20/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: */24 */24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 */24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: */24 */24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 17/24 7/24 0/24 0/24 
agent 2: 1/24 5/24 18/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 */24 8/24 
agent 3: 1/24 5/24 */24 8/24 
agent 4: */24 */24 10/24 8/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 */24 8/24 
agent 3: 1/24 5/24 */24 8/24 
agent 4: */24 */24 10/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 */24 8/24 
agent 3: 1/24 5/24 10/24 8/24 
agent 4: */24 */24 10/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: */24 6/24 4/24 8/24 
agent 3: 1/24 5/24 10/24 8/24 
agent 4: */24 */24 10/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 1/24 5/24 10/24 8/24 
agent 4: */24 */24 10/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 6/24 */24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 */24 */24 
agent 4: 0/24 3/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 10/24 11/24 
agent 4: 0/24 3/24 10/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 2/24 
agent 3: 0/24 1/24 12/24 11/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 */24 11/24 
agent 4: 0/24 3/24 10/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 2/24 
agent 3: 0/24 1/24 12/24 11/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 0/24 2/24 
agent 3: 0/24 1/24 12/24 11/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 0/24 2/24 
agent 3: 0/24 1/24 12/24 11/24 
agent 4: 0/24 1/24 12/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 6/24 10/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 6/24 10/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: */24 */24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 6/24 10/24 0/24 8/24 
agent 3: 1/24 3/24 12/24 8/24 
agent 4: 1/24 3/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 */24 11/24 
agent 4: 0/24 3/24 10/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 10/24 11/24 
agent 4: 0/24 3/24 10/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 */24 
agent 3: 0/24 3/24 10/24 11/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 1/24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 1/24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 1/24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 1/24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 1/24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 18/24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 1/24 */24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 */24 */24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 1/24 9/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 2/24 
agent 3: 2/24 1/24 10/24 11/24 
agent 4: 0/24 */24 */24 11/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 1/24 23/24 0/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 */24 */24 5/24 
agent 4: 0/24 1/24 */24 14/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 */24 5/24 
agent 4: 0/24 1/24 9/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 7/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 7/24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 5/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 5/24 19/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 7/24 17/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 2/24 
agent 3: 0/24 10/24 3/24 */24 
agent 4: 0/24 2/24 11/24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 5/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 2/24 
agent 3: 0/24 10/24 3/24 */24 
agent 4: 0/24 2/24 11/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 2/24 
agent 3: 0/24 10/24 3/24 11/24 
agent 4: 0/24 2/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 8/24 2/24 
agent 3: 2/24 */24 3/24 11/24 
agent 4: 0/24 2/24 11/24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 */24 11/24 
agent 4: 0/24 2/24 11/24 11/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 */24 4/24 11/24 
agent 4: 0/24 */24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 8/24 2/24 
agent 3: 2/24 */24 3/24 11/24 
agent 4: 0/24 2/24 11/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 8/24 2/24 
agent 3: 2/24 8/24 3/24 11/24 
agent 4: 0/24 2/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 1/24 */24 5/24 
agent 4: 0/24 1/24 9/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 1/24 */24 5/24 
agent 4: 0/24 1/24 9/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 1/24 9/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 2/24 
agent 3: 2/24 1/24 10/24 11/24 
agent 4: 0/24 1/24 */24 11/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 1/24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 2/24 
agent 3: 2/24 1/24 10/24 11/24 
agent 4: 0/24 1/24 */24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 2/24 
agent 3: 2/24 1/24 10/24 11/24 
agent 4: 0/24 1/24 12/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 0/24 2/24 
agent 3: 0/24 1/24 12/24 11/24 
agent 4: 0/24 */24 12/24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 2/24 
agent 3: 2/24 1/24 10/24 11/24 
agent 4: 0/24 */24 12/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 0/24 2/24 
agent 3: 0/24 1/24 12/24 11/24 
agent 4: 0/24 */24 12/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 0/24 2/24 
agent 3: 0/24 1/24 12/24 11/24 
agent 4: 0/24 1/24 12/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 */24 12/24 11/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 */24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 */24 
agent 3: 0/24 1/24 12/24 11/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: 0/24 */24 12/24 9/24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 0/24 9/24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 1/24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 */24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 1/24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 1/24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 1/24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 */24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 2/24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > a > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 2/24 0/24 
agent 2: 11/24 3/24 10/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 */24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 */24 12/24 */24 
agent 4: 0/24 */24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 */24 2/24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: */24 11/24 */24 */24 
agent 4: 0/24 11/24 3/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 9/24 2/24 
agent 2: 11/24 0/24 3/24 10/24 
agent 3: 2/24 11/24 9/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 9/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: 2/24 11/24 0/24 */24 
agent 4: 0/24 11/24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 9/24 2/24 
agent 2: 8/24 */24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 10/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 0/24 
agent 2: 11/24 0/24 */24 8/24 
agent 3: 2/24 11/24 */24 */24 
agent 4: 0/24 11/24 */24 9/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 9/24 2/24 
agent 2: 8/24 */24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 2/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 */24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 4/24 
agent 2: 8/24 5/24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 4/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 0/24 
agent 2: 8/24 5/24 */24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 2/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 */24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 */24 0/24 
agent 2: 8/24 1/24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 1/24 0/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 */24 0/24 
agent 2: 8/24 1/24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 */24 18/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 18/24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 15/24 
agent 4: 0/24 */24 18/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 */24 0/24 
agent 2: 8/24 1/24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 1/24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 2/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 4/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 2/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 2/24 12/24 6/24 4/24 
agent 4: 0/24 0/24 4/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 */24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 16/24 8/24 0/24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 */24 4/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 1/24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 */24 1/24 20/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 */24 1/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 */24 1/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 */24 1/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 */24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 3/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 1/24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 */24 18/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 */24 23/24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 1/24 */24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 1/24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 */24 18/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 17/24 1/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 1/24 15/24 0/24 
agent 3: 8/24 7/24 3/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 1/24 */24 5/24 
agent 4: 0/24 1/24 */24 14/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 17/24 1/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 */24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 */24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > a > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 */24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 */24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 1/24 0/24 
agent 2: 11/24 10/24 1/24 2/24 
agent 3: 2/24 1/24 10/24 11/24 
agent 4: 0/24 1/24 12/24 11/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 1/24 23/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 1/24 23/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 */24 0/24 13/24 
agent 4: 0/24 */24 18/24 4/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 17/24 1/24 0/24 
agent 2: 6/24 1/24 */24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 */24 23/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 */24 0/24 13/24 
agent 4: 0/24 */24 18/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 */24 */24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 18/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 18/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: */24 3/24 12/24 7/24 
agent 4: */24 3/24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 18/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 */24 12/24 4/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: */24 3/24 12/24 7/24 
agent 4: */24 3/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: */24 3/24 12/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: */24 3/24 12/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: 2/24 */24 12/24 3/24 
agent 4: */24 3/24 12/24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 2/24 0/24 12/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 2/24 3/24 12/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 18/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: */24 7/24 12/24 3/24 
agent 4: 2/24 */24 12/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 17/24 0/24 1/24 
agent 2: 6/24 4/24 12/24 2/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 1/24 0/24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: */24 7/24 12/24 3/24 
agent 4: 2/24 */24 12/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 0/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 2/24 7/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 8/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: */24 */24 8/24 3/24 
agent 4: 2/24 11/24 8/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 8/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: */24 */24 8/24 3/24 
agent 4: 2/24 11/24 8/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: */24 */24 8/24 3/24 
agent 4: 2/24 11/24 8/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 2/24 11/24 8/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 10/24 
agent 4: */24 11/24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 */24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 10/24 
agent 4: */24 11/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: */24 0/24 */24 */24 
agent 3: */24 */24 1/24 10/24 
agent 4: */24 11/24 1/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: */24 0/24 12/24 */24 
agent 3: */24 11/24 1/24 10/24 
agent 4: */24 11/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: */24 0/24 12/24 */24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: */24 11/24 1/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 */24 2/24 
agent 2: */24 0/24 12/24 */24 
agent 3: 2/24 11/24 1/24 */24 
agent 4: */24 11/24 1/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 */24 4/24 
agent 2: 10/24 0/24 10/24 4/24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 2/24 11/24 0/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 */24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 4/24 
agent 2: 8/24 4/24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 12/24 8/24 4/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 2/24 2/24 */24 
agent 3: 2/24 10/24 2/24 10/24 
agent 4: 0/24 10/24 12/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 */24 4/24 
agent 2: 8/24 4/24 */24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 12/24 8/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 8/24 4/24 
agent 2: 8/24 4/24 8/24 4/24 
agent 3: 8/24 4/24 0/24 12/24 
agent 4: 0/24 12/24 8/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 0/24 10/24 4/24 
agent 3: 2/24 11/24 6/24 5/24 
agent 4: 2/24 11/24 0/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 */24 8/24 4/24 
agent 4: 0/24 10/24 10/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > c > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 */24 8/24 4/24 
agent 4: 0/24 10/24 10/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 6/24 5/24 
agent 2: 11/24 2/24 0/24 11/24 
agent 3: 2/24 10/24 8/24 4/24 
agent 4: 0/24 10/24 10/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 10/24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 10/24 */24 4/24 
agent 4: 0/24 */24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 */24 */24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 10/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 */24 0/24 10/24 
agent 3: 0/24 10/24 10/24 */24 
agent 4: 0/24 10/24 */24 4/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 3/24 7/24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 10/24 */24 4/24 
agent 4: 0/24 10/24 10/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 */24 0/24 10/24 
agent 3: 0/24 10/24 10/24 */24 
agent 4: 0/24 10/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 10/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 10/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 10/24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 18/24 0/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 0/24 */24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 11/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 2/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 10/24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 10/24 */24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 18/24 0/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 10/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 18/24 0/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 6/24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 6/24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 4/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: 0/24 */24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: 5/24 8/24 2/24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 22/24 0/24 2/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 5/24 2/24 8/24 9/24 
agent 3: */24 2/24 8/24 */24 
agent 4: 0/24 */24 8/24 2/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 22/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 22/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 6/24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 4/24 0/24 
agent 2: 5/24 2/24 9/24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 2/24 8/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 22/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 22/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 6/24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 */24 */24 11/24 
agent 4: 0/24 */24 */24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 9/24 4/24 11/24 
agent 4: 0/24 9/24 4/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 */24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 11/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 4/24 */24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 9/24 4/24 11/24 
agent 4: 0/24 */24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 */24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 2/24 10/24 
agent 3: 0/24 */24 */24 4/24 
agent 4: 0/24 */24 11/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 2/24 10/24 
agent 3: 0/24 9/24 11/24 4/24 
agent 4: 0/24 9/24 11/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 2/24 6/24 
agent 3: 0/24 7/24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 2/24 10/24 
agent 3: 0/24 9/24 11/24 4/24 
agent 4: 0/24 9/24 11/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 */24 2/24 6/24 
agent 3: 0/24 7/24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 7/24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 */24 
agent 2: 6/24 4/24 8/24 6/24 
agent 3: 0/24 9/24 8/24 7/24 
agent 4: 0/24 */24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 4/24 */24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 4/24 */24 
agent 4: 0/24 */24 4/24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 */24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 */24 9/24 
agent 4: 0/24 11/24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 */24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 */24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 4/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 2/24 */24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 6/24 */24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 */24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 6/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 2/24 0/24 
agent 2: 8/24 2/24 */24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 6/24 */24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 2/24 9/24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 6/24 4/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 4/24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 */24 4/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 */24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: */24 0/24 12/24 */24 
agent 3: 2/24 11/24 1/24 */24 
agent 4: */24 11/24 1/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: */24 0/24 12/24 */24 
agent 3: 2/24 11/24 1/24 */24 
agent 4: */24 11/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: */24 0/24 12/24 */24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: */24 11/24 1/24 */24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 6/24 */24 1/24 9/24 
agent 4: 6/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 6/24 */24 1/24 9/24 
agent 4: 6/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: */24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > a > b > d
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 */24 
agent 3: 8/24 */24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 0/24 9/24 7/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 18/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 */24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 22/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: 5/24 8/24 2/24 9/24 
agent 3: 5/24 8/24 2/24 9/24 
agent 4: 0/24 8/24 */24 2/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 22/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 22/24 0/24 2/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 7/24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 22/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 */24 
agent 2: 5/24 2/24 8/24 9/24 
agent 3: 5/24 2/24 8/24 9/24 
agent 4: 0/24 */24 8/24 2/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 22/24 0/24 2/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 7/24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 7/24 11/24 6/24 
agent 4: 0/24 */24 11/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 7/24 11/24 6/24 
agent 4: 0/24 7/24 11/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 2/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 10/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 */24 */24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 2/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 10/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 */24 */24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 10/24 8/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 */24 */24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 10/24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 9/24 7/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 18/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 10/24 8/24 6/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 9/24 7/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 18/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 9/24 7/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: 6/24 0/24 */24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > c > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: 6/24 0/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: */24 0/24 */24 */24 
agent 4: 6/24 0/24 10/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 */24 */24 
agent 4: 6/24 0/24 10/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 4/24 
agent 3: 6/24 */24 8/24 */24 
agent 4: 6/24 1/24 8/24 9/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 11/24 2/24 5/24 
agent 2: 6/24 11/24 2/24 5/24 
agent 3: 6/24 2/24 10/24 6/24 
agent 4: */24 0/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > c > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: 6/24 */24 8/24 4/24 
agent 3: 6/24 */24 8/24 */24 
agent 4: 6/24 1/24 8/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 0/24 */24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 6/24 */24 8/24 */24 
agent 4: 6/24 1/24 8/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 6/24 14/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 11/24 2/24 5/24 
agent 2: 6/24 11/24 2/24 5/24 
agent 3: 6/24 2/24 10/24 6/24 
agent 4: */24 0/24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 11/24 2/24 5/24 
agent 2: 6/24 11/24 2/24 5/24 
agent 3: 6/24 2/24 10/24 6/24 
agent 4: 6/24 0/24 10/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 10/24 */24 
agent 4: 6/24 0/24 10/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 16/24 2/24 
agent 4: 6/24 0/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 6/24 
agent 2: 6/24 10/24 */24 6/24 
agent 3: 6/24 */24 */24 6/24 
agent 4: 6/24 2/24 10/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > c > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 10/24 */24 
agent 4: 6/24 0/24 10/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 */24 */24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 10/24 8/24 
agent 4: 6/24 0/24 10/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 2/24 4/24 
agent 2: 6/24 12/24 2/24 4/24 
agent 3: 6/24 0/24 10/24 8/24 
agent 4: 6/24 0/24 10/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 2/24 4/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 0/24 8/24 9/24 */24 
agent 4: 0/24 8/24 9/24 7/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 4/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 2/24 4/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 0/24 8/24 9/24 */24 
agent 4: 0/24 8/24 9/24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 2/24 4/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 0/24 8/24 9/24 7/24 
agent 4: 0/24 8/24 9/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 2/24 4/24 
agent 2: 6/24 8/24 4/24 6/24 
agent 3: 0/24 8/24 9/24 7/24 
agent 4: 0/24 8/24 9/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 */24 0/24 4/24 
agent 2: 6/24 4/24 8/24 6/24 
agent 3: 0/24 9/24 8/24 7/24 
agent 4: 0/24 */24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 10/24 8/24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 6/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 */24 0/24 4/24 
agent 2: 6/24 4/24 8/24 6/24 
agent 3: 0/24 9/24 8/24 7/24 
agent 4: 0/24 */24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 2/24 0/24 4/24 
agent 2: 6/24 4/24 8/24 6/24 
agent 3: 0/24 9/24 8/24 7/24 
agent 4: 0/24 */24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 2/24 0/24 4/24 
agent 2: 6/24 4/24 8/24 6/24 
agent 3: 0/24 9/24 8/24 7/24 
agent 4: 0/24 9/24 8/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 4/24 
agent 2: 5/24 2/24 8/24 9/24 
agent 3: 5/24 2/24 8/24 9/24 
agent 4: 0/24 */24 8/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 2/24 */24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: 0/24 9/24 7/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: */24 8/24 8/24 6/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 0/24 4/24 
agent 2: 5/24 2/24 8/24 9/24 
agent 3: 5/24 2/24 8/24 9/24 
agent 4: 0/24 */24 8/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 6/24 0/24 4/24 
agent 2: 5/24 2/24 8/24 9/24 
agent 3: 5/24 2/24 8/24 9/24 
agent 4: 0/24 14/24 8/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 4/24 
agent 2: 5/24 8/24 2/24 9/24 
agent 3: 5/24 8/24 2/24 9/24 
agent 4: 0/24 8/24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 14/24 6/24 */24 0/24 
agent 2: 5/24 2/24 9/24 8/24 
agent 3: 5/24 2/24 9/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 4/24 */24 8/24 2/24 
agent 3: */24 2/24 8/24 10/24 
agent 4: */24 2/24 8/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 4/24 
agent 2: 5/24 2/24 8/24 9/24 
agent 3: */24 2/24 8/24 */24 
agent 4: 0/24 14/24 8/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 0/24 */24 4/24 
agent 2: 5/24 8/24 2/24 9/24 
agent 3: 5/24 8/24 2/24 9/24 
agent 4: 0/24 8/24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 6/24 4/24 
agent 2: 5/24 8/24 2/24 9/24 
agent 3: 5/24 8/24 2/24 9/24 
agent 4: 0/24 8/24 14/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 6/24 4/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 */24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 0/24 */24 */24 
agent 2: 4/24 8/24 */24 2/24 
agent 3: 4/24 8/24 2/24 10/24 
agent 4: */24 8/24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 */24 */24 
agent 2: 5/24 8/24 2/24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 14/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 8/24 6/24 4/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 6/24 4/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 16/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 6/24 4/24 
agent 2: 6/24 8/24 1/24 9/24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 16/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > b > d > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 6/24 */24 */24 
agent 2: 6/24 6/24 */24 2/24 
agent 3: 6/24 6/24 */24 */24 
agent 4: */24 */24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 */24 4/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 16/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > b > d > c
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 6/24 */24 */24 
agent 2: 6/24 6/24 10/24 2/24 
agent 3: 6/24 6/24 */24 */24 
agent 4: */24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 6/24 */24 */24 
agent 2: 6/24 6/24 10/24 2/24 
agent 3: 6/24 6/24 */24 */24 
agent 4: 6/24 6/24 */24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 */24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 10/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > c > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 8/24 */24 4/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 16/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 6/24 4/24 
agent 2: 6/24 8/24 */24 */24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 16/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 6/24 4/24 
agent 2: 6/24 8/24 1/24 9/24 
agent 3: 6/24 8/24 1/24 9/24 
agent 4: 6/24 0/24 16/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 0/24 6/24 4/24 
agent 2: 5/24 8/24 2/24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 14/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 0/24 6/24 4/24 
agent 2: 5/24 8/24 2/24 */24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 14/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 6/24 4/24 
agent 2: 5/24 8/24 2/24 9/24 
agent 3: */24 8/24 2/24 */24 
agent 4: 0/24 8/24 14/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 6/24 4/24 
agent 2: 5/24 8/24 2/24 9/24 
agent 3: 5/24 8/24 2/24 9/24 
agent 4: 0/24 8/24 14/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 */24 0/24 4/24 
agent 2: 5/24 2/24 8/24 9/24 
agent 3: */24 2/24 8/24 */24 
agent 4: 0/24 14/24 8/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 2/24 2/24 9/24 
agent 3: 2/24 10/24 2/24 10/24 
agent 4: 0/24 10/24 12/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 */24 0/24 4/24 
agent 2: 5/24 2/24 8/24 9/24 
agent 3: */24 2/24 8/24 */24 
agent 4: 0/24 14/24 8/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 6/24 0/24 4/24 
agent 2: 5/24 2/24 8/24 9/24 
agent 3: */24 2/24 8/24 */24 
agent 4: 0/24 14/24 8/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 6/24 0/24 4/24 
agent 2: 5/24 2/24 8/24 9/24 
agent 3: 5/24 2/24 8/24 9/24 
agent 4: 0/24 14/24 8/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 14/24 6/24 4/24 0/24 
agent 2: 5/24 2/24 9/24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 2/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 6/24 4/24 0/24 
agent 2: 5/24 2/24 9/24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 */24 2/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 6/24 4/24 0/24 
agent 2: 5/24 2/24 9/24 8/24 
agent 3: */24 2/24 */24 8/24 
agent 4: 0/24 14/24 2/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 6/24 4/24 0/24 
agent 2: 5/24 2/24 9/24 8/24 
agent 3: 5/24 2/24 9/24 8/24 
agent 4: 0/24 14/24 2/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 14/24 6/24 */24 0/24 
agent 2: 5/24 2/24 9/24 8/24 
agent 3: 5/24 2/24 9/24 8/24 
agent 4: 0/24 14/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > a > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 6/24 */24 0/24 
agent 2: 5/24 2/24 9/24 8/24 
agent 3: 5/24 2/24 9/24 8/24 
agent 4: 0/24 14/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 6/24 4/24 0/24 
agent 2: 5/24 2/24 9/24 8/24 
agent 3: 5/24 2/24 9/24 8/24 
agent 4: 0/24 14/24 2/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 2/24 4/24 0/24 
agent 2: 6/24 */24 */24 8/24 
agent 3: 0/24 9/24 7/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 4/24 */24 2/24 8/24 
agent 3: 4/24 2/24 10/24 8/24 
agent 4: */24 2/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 */24 2/24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 */24 */24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 6/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 */24 
agent 2: 0/24 22/24 0/24 2/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: */24 12/24 2/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 22/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 6/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 */24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 22/24 0/24 2/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 2/24 0/24 16/24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 2/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 9/24 7/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 6/24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 17/24 7/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 9/24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 17/24 */24 5/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 4/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 6/24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 */24 2/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 */24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 10/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 2/24 10/24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 10/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 10/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 10/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 10/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 2/24 10/24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 10/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: */24 0/24 12/24 2/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: */24 11/24 1/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 22/24 0/24 2/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 22/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: */24 12/24 0/24 2/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: */24 0/24 12/24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 22/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 */24 */24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 2/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 9/24 7/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 7/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 9/24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 17/24 */24 5/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 */24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 17/24 7/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 */24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 1/24 9/24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 17/24 */24 5/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 */24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 19/24 */24 1/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 */24 
agent 3: 0/24 19/24 */24 1/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 */24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 9/24 4/24 11/24 
agent 4: 0/24 */24 4/24 11/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 */24 9/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 4/24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 4/24 */24 
agent 4: 0/24 1/24 4/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 4/24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 1/24 4/24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 5/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 1/24 4/24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 5/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 4/24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 5/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 */24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 1/24 9/24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 17/24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 0/24 
agent 2: 8/24 5/24 4/24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 */24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 */24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 */24 */24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 */24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 6/24 16/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 4/24 9/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 0/24 
agent 2: 11/24 0/24 */24 8/24 
agent 3: 2/24 11/24 4/24 */24 
agent 4: 0/24 11/24 */24 9/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 */24 
agent 4: 0/24 1/24 4/24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 4/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 */24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 9/24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 17/24 */24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 9/24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 17/24 */24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 9/24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 17/24 2/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 9/24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 17/24 2/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 9/24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 17/24 2/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 9/24 6/24 
agent 3: 8/24 0/24 9/24 7/24 
agent 4: 0/24 17/24 2/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 4/24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 17/24 1/24 6/24 
agent 4: 0/24 1/24 17/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 */24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 6/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 4/24 19/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 */24 4/24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 4/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 5/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 4/24 */24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 5/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 4/24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 17/24 5/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 11/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 4/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 11/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 4/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 5/24 11/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 4/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 5/24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 18/24 4/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 11/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 */24 11/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 5/24 11/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 18/24 4/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 0/24 5/24 11/24 
agent 3: 8/24 0/24 5/24 11/24 
agent 4: 0/24 18/24 4/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 5/24 19/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 6/24 0/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 */24 */24 
agent 4: 6/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 5/24 19/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 */24 17/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 */24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 7/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 6/24 0/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 */24 */24 
agent 4: 6/24 0/24 7/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 12/24 6/24 0/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 */24 */24 
agent 4: 6/24 0/24 7/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 6/24 0/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 */24 */24 
agent 4: 6/24 0/24 7/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 6/24 0/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 7/24 11/24 
agent 4: 6/24 0/24 7/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 */24 4/24 8/24 
agent 3: 6/24 */24 */24 8/24 
agent 4: 6/24 */24 */24 8/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 */24 5/24 2/24 
agent 3: 6/24 2/24 5/24 11/24 
agent 4: 6/24 0/24 7/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 */24 4/24 8/24 
agent 3: 6/24 */24 */24 8/24 
agent 4: 6/24 */24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 */24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 6/24 */24 */24 8/24 
agent 4: 6/24 */24 */24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 6/24 4/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 */24 5/24 2/24 
agent 3: 6/24 2/24 5/24 11/24 
agent 4: 6/24 0/24 7/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 11/24 5/24 2/24 
agent 3: 6/24 2/24 5/24 11/24 
agent 4: 6/24 0/24 7/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 11/24 7/24 0/24 
agent 3: 6/24 2/24 */24 12/24 
agent 4: 6/24 0/24 6/24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 6/24 */24 
agent 3: 6/24 */24 5/24 */24 
agent 4: 6/24 2/24 5/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 11/24 7/24 0/24 
agent 3: 6/24 2/24 */24 12/24 
agent 4: 6/24 0/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 11/24 7/24 0/24 
agent 2: 6/24 11/24 7/24 0/24 
agent 3: 6/24 2/24 4/24 12/24 
agent 4: 6/24 0/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 5/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 2/24 4/24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: 6/24 8/24 4/24 */24 
agent 4: 6/24 0/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 5/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 2/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 5/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 2/24 4/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 5/24 0/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 10/24 4/24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 2/24 0/24 10/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 */24 0/24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 */24 4/24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 0/24 6/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 10/24 4/24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 2/24 0/24 10/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 10/24 4/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 2/24 0/24 10/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 0/24 6/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 12/24 */24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 0/24 6/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 6/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 0/24 6/24 18/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 0/24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: 6/24 8/24 4/24 */24 
agent 4: 6/24 0/24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 14/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 14/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 4/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 14/24 
agent 4: 0/24 14/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: 6/24 8/24 4/24 6/24 
agent 4: 6/24 0/24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 14/24 
agent 4: 0/24 14/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 4/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 2/24 */24 
agent 4: 0/24 12/24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: 6/24 8/24 4/24 6/24 
agent 4: 6/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 */24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 */24 18/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: 6/24 8/24 4/24 6/24 
agent 4: 6/24 0/24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 6/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 */24 */24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 */24 0/24 
agent 2: 10/24 8/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 2/24 */24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 10/24 8/24 0/24 
agent 2: 6/24 */24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 2/24 4/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 */24 4/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 8/24 */24 0/24 
agent 2: 10/24 8/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 2/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: 10/24 8/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 2/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: 10/24 8/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 6/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 */24 4/24 18/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 */24 */24 12/24 
agent 4: 0/24 6/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 */24 4/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 */24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 */24 0/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 4/24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 */24 */24 
agent 4: 0/24 2/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 */24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 2/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 2/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: 2/24 */24 2/24 10/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 */24 9/24 2/24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 14/24 4/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 2/24 */24 
agent 4: 0/24 12/24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 */24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: 2/24 */24 2/24 10/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: 2/24 10/24 2/24 10/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: 2/24 10/24 2/24 10/24 
agent 4: 0/24 10/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 10/24 0/24 
agent 3: 0/24 10/24 2/24 12/24 
agent 4: 0/24 10/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 8/24 3/24 
agent 2: 11/24 2/24 8/24 3/24 
agent 3: 2/24 10/24 8/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 */24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 */24 */24 
agent 4: */24 10/24 2/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: 2/24 10/24 2/24 */24 
agent 4: 0/24 10/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 10/24 0/24 
agent 3: 0/24 10/24 2/24 12/24 
agent 4: 0/24 10/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 2/24 10/24 0/24 
agent 3: 0/24 10/24 2/24 12/24 
agent 4: 0/24 10/24 2/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 2/24 10/24 0/24 
agent 3: 0/24 10/24 2/24 12/24 
agent 4: 0/24 10/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 10/24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 10/24 */24 4/24 
agent 4: 0/24 10/24 10/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 10/24 0/24 
agent 2: 8/24 */24 3/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 17/24 7/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 10/24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 10/24 */24 4/24 
agent 4: 0/24 10/24 10/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 10/24 */24 4/24 
agent 4: 0/24 10/24 10/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 0/24 10/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 10/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 2/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 6/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 */24 6/24 12/24 
agent 4: 0/24 6/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 6/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 6/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: 2/24 10/24 2/24 */24 
agent 4: 0/24 10/24 2/24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 2/24 */24 18/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 */24 */24 
agent 4: 0/24 12/24 2/24 10/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 2/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 */24 6/24 */24 
agent 4: 0/24 2/24 4/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 9/24 7/24 0/24 
agent 2: 8/24 9/24 7/24 0/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 6/24 */24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 4/24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 0/24 */24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 6/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 6/24 0/24 
agent 2: 11/24 7/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 14/24 4/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 14/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 4/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 2/24 */24 
agent 4: 0/24 12/24 2/24 10/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 14/24 
agent 4: 0/24 14/24 */24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 */24 9/24 2/24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 14/24 4/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 2/24 */24 
agent 4: 0/24 12/24 2/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 2/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 2/24 10/24 1/24 
agent 3: 2/24 10/24 2/24 10/24 
agent 4: 0/24 10/24 2/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 14/24 
agent 4: 0/24 14/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 4/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 */24 6/24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 */24 */24 
agent 4: 0/24 14/24 4/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 11/24 9/24 4/24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 11/24 9/24 4/24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 4/24 0/24 
agent 2: 11/24 9/24 4/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 4/24 0/24 
agent 2: 11/24 9/24 4/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 9/24 0/24 4/24 
agent 3: 2/24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 */24 */24 0/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 9/24 0/24 4/24 
agent 3: 2/24 0/24 12/24 */24 
agent 4: 0/24 6/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 9/24 0/24 4/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 4/24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: 2/24 11/24 0/24 */24 
agent 4: 0/24 11/24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 */24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 2/24 17/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 */24 4/24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: 2/24 11/24 0/24 */24 
agent 4: 0/24 11/24 8/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 7/24 4/24 
agent 2: 11/24 0/24 9/24 4/24 
agent 3: 2/24 11/24 0/24 11/24 
agent 4: 0/24 11/24 8/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 7/24 4/24 
agent 2: 8/24 5/24 0/24 */24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 7/24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 */24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 7/24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 7/24 4/24 
agent 2: 8/24 5/24 0/24 11/24 
agent 3: 8/24 0/24 12/24 4/24 
agent 4: 0/24 14/24 5/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 */24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 2/24 17/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 */24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 2/24 17/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 1/24 4/24 
agent 2: 8/24 11/24 0/24 5/24 
agent 3: 8/24 0/24 6/24 10/24 
agent 4: 0/24 2/24 17/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 1/24 18/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 1/24 18/24 5/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > d > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 0/24 8/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 */24 18/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 */24 18/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 9/24 */24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 1/24 18/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 9/24 */24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 9/24 3/24 
agent 3: 0/24 11/24 8/24 5/24 
agent 4: 0/24 11/24 0/24 13/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 9/24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 */24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 */24 
agent 2: 11/24 0/24 12/24 1/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 11/24 1/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 */24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 */24 
agent 2: 11/24 12/24 0/24 1/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 3/24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 3/24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 9/24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 9/24 0/24 */24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 9/24 0/24 3/24 
agent 3: 0/24 6/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 */24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 */24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: 11/24 12/24 0/24 1/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 0/24 1/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 */24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 */24 22/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 8/24 0/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 */24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 */24 2/24 
agent 4: 0/24 0/24 2/24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 14/24 4/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 */24 
agent 4: 0/24 14/24 4/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 4/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 */24 9/24 2/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 4/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 */24 9/24 2/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 4/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 9/24 2/24 
agent 2: 8/24 5/24 9/24 2/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 4/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 0/24 5/24 8/24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: 0/24 11/24 4/24 9/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 10/24 0/24 
agent 2: 8/24 */24 4/24 11/24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 4/24 11/24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 4/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 4/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 9/24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 */24 4/24 */24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 4/24 11/24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 6/24 2/24 
agent 3: 6/24 */24 5/24 */24 
agent 4: 6/24 2/24 5/24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 16/24 */24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 */24 6/24 2/24 
agent 3: 6/24 */24 5/24 */24 
agent 4: 6/24 2/24 5/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 */24 0/24 
agent 2: 6/24 10/24 6/24 2/24 
agent 3: 6/24 */24 5/24 */24 
agent 4: 6/24 2/24 5/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 */24 8/24 0/24 
agent 2: 6/24 10/24 6/24 2/24 
agent 3: 6/24 */24 5/24 11/24 
agent 4: 6/24 2/24 5/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 10/24 8/24 0/24 
agent 2: 6/24 10/24 6/24 2/24 
agent 3: 6/24 2/24 5/24 11/24 
agent 4: 6/24 2/24 5/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: 6/24 8/24 6/24 4/24 
agent 3: */24 8/24 5/24 */24 
agent 4: 1/24 8/24 5/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: 6/24 */24 4/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 10/24 */24 6/24 
agent 2: 6/24 10/24 */24 6/24 
agent 3: 6/24 */24 */24 6/24 
agent 4: 6/24 2/24 10/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 10/24 8/24 0/24 
agent 2: 6/24 10/24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 2/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 8/24 0/24 
agent 2: 6/24 8/24 6/24 4/24 
agent 3: */24 8/24 5/24 */24 
agent 4: 1/24 8/24 5/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 6/24 8/24 6/24 4/24 
agent 3: */24 8/24 5/24 */24 
agent 4: 1/24 8/24 5/24 10/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 8/24 0/24 
agent 2: 6/24 8/24 6/24 4/24 
agent 3: 1/24 8/24 5/24 10/24 
agent 4: 1/24 8/24 5/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 1/24 5/24 8/24 10/24 
agent 4: 1/24 5/24 8/24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 6/24 8/24 */24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 1/24 8/24 9/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > a > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 0/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 1/24 5/24 8/24 10/24 
agent 4: 1/24 5/24 8/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 1/24 5/24 8/24 10/24 
agent 4: 1/24 5/24 8/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 1/24 5/24 10/24 8/24 
agent 4: */24 */24 10/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 1/24 5/24 10/24 8/24 
agent 4: */24 */24 10/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 1/24 5/24 10/24 8/24 
agent 4: 1/24 5/24 10/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 6/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: 1/24 9/24 6/24 8/24 
agent 4: */24 9/24 6/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 6/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: 1/24 9/24 6/24 8/24 
agent 4: */24 9/24 6/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: */24 */24 6/24 8/24 
agent 3: 1/24 9/24 6/24 8/24 
agent 4: 1/24 9/24 6/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 6/24 4/24 6/24 8/24 
agent 3: 1/24 9/24 6/24 8/24 
agent 4: 1/24 9/24 6/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: */24 */24 8/24 6/24 
agent 3: 1/24 */24 8/24 6/24 
agent 4: */24 9/24 8/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 6/24 0/24 */24 8/24 
agent 3: 1/24 */24 4/24 8/24 
agent 4: */24 11/24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: */24 */24 8/24 6/24 
agent 3: 1/24 */24 8/24 6/24 
agent 4: */24 9/24 8/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: */24 */24 8/24 6/24 
agent 3: 1/24 9/24 8/24 6/24 
agent 4: 1/24 9/24 8/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 6/24 4/24 8/24 6/24 
agent 3: 1/24 9/24 8/24 6/24 
agent 4: 1/24 9/24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 7/24 11/24 6/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 6/24 0/24 8/24 */24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: */24 11/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 7/24 11/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 7/24 11/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 7/24 11/24 6/24 
agent 4: 0/24 7/24 11/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 2/24 */24 
agent 3: 0/24 9/24 11/24 4/24 
agent 4: 0/24 9/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 2/24 */24 
agent 3: 0/24 9/24 11/24 4/24 
agent 4: 0/24 9/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 2/24 10/24 
agent 3: 0/24 9/24 11/24 4/24 
agent 4: 0/24 9/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 2/24 10/24 
agent 3: 0/24 9/24 11/24 4/24 
agent 4: 0/24 9/24 11/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 6/24 0/24 8/24 10/24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: */24 11/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 6/24 0/24 8/24 10/24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: */24 11/24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 6/24 0/24 8/24 10/24 
agent 3: 1/24 11/24 8/24 4/24 
agent 4: 1/24 11/24 8/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 4/24 6/24 */24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 0/24 */24 2/24 
agent 3: 0/24 */24 4/24 11/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 0/24 */24 2/24 
agent 3: 0/24 */24 4/24 11/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 9/24 4/24 11/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 9/24 4/24 11/24 
agent 4: 0/24 9/24 4/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 6/24 0/24 10/24 8/24 
agent 3: 1/24 */24 4/24 8/24 
agent 4: */24 11/24 */24 8/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 */24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 */24 4/24 11/24 
agent 4: 0/24 9/24 4/24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 10/24 2/24 
agent 3: 0/24 9/24 4/24 11/24 
agent 4: 0/24 9/24 4/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: */24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: */24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 5/24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 5/24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 */24 4/24 11/24 
agent 4: 0/24 9/24 4/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 8/24 5/24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 */24 4/24 11/24 
agent 4: 0/24 9/24 4/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 8/24 5/24 0/24 
agent 2: 11/24 0/24 11/24 2/24 
agent 3: 2/24 7/24 4/24 11/24 
agent 4: 0/24 9/24 4/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 7/24 4/24 5/24 
agent 3: 8/24 0/24 11/24 5/24 
agent 4: 0/24 5/24 5/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: */24 4/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 0/24 12/24 */24 
agent 3: */24 7/24 4/24 */24 
agent 4: 2/24 */24 4/24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 0/24 12/24 */24 
agent 3: */24 7/24 4/24 */24 
agent 4: 2/24 */24 4/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 0/24 12/24 2/24 
agent 3: */24 7/24 4/24 */24 
agent 4: 2/24 7/24 4/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 10/24 4/24 0/24 
agent 2: 10/24 0/24 12/24 2/24 
agent 3: 2/24 7/24 4/24 11/24 
agent 4: 2/24 7/24 4/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 */24 0/24 
agent 2: 10/24 */24 */24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: */24 */24 */24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: 10/24 6/24 6/24 2/24 
agent 3: */24 5/24 6/24 */24 
agent 4: 2/24 5/24 6/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 */24 0/24 
agent 2: 10/24 */24 */24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: */24 */24 */24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 */24 */24 0/24 
agent 2: 10/24 */24 */24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 2/24 */24 */24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: */24 1/24 11/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 12/24 2/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: */24 0/24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 0/24 */24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: */24 1/24 11/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 */24 0/24 */24 
agent 2: */24 */24 2/24 */24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: */24 1/24 11/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 2/24 8/24 
agent 3: 2/24 3/24 11/24 8/24 
agent 4: 2/24 3/24 11/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: */24 4/24 2/24 8/24 
agent 3: 2/24 3/24 11/24 8/24 
agent 4: 2/24 3/24 11/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 2/24 8/24 
agent 3: 2/24 3/24 11/24 8/24 
agent 4: 2/24 3/24 11/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: */24 3/24 12/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 0/24 */24 
agent 2: 10/24 */24 2/24 */24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: */24 1/24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 0/24 */24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: */24 3/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 0/24 10/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: */24 3/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 0/24 10/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 2/24 3/24 12/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 0/24 10/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: */24 3/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 0/24 10/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: */24 3/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 14/24 0/24 0/24 
agent 2: 10/24 4/24 0/24 10/24 
agent 3: 2/24 3/24 12/24 7/24 
agent 4: 2/24 3/24 12/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 12/24 0/24 2/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 12/24 0/24 2/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 12/24 0/24 2/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 2/24 0/24 12/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 12/24 0/24 */24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: */24 0/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 12/24 0/24 */24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: */24 0/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 12/24 0/24 2/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: */24 0/24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 12/24 0/24 2/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 2/24 0/24 12/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 */24 0/24 */24 
agent 2: 10/24 */24 2/24 2/24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: */24 1/24 11/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 */24 0/24 */24 
agent 2: 10/24 */24 2/24 2/24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: */24 1/24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 */24 0/24 */24 
agent 2: 10/24 10/24 2/24 2/24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: */24 1/24 11/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 0/24 */24 
agent 2: 10/24 10/24 2/24 2/24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: 2/24 1/24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 0/24 2/24 
agent 2: 10/24 10/24 2/24 2/24 
agent 3: 2/24 1/24 11/24 10/24 
agent 4: 2/24 1/24 11/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 2/24 
agent 2: */24 16/24 0/24 6/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 */24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 */24 */24 
agent 4: */24 10/24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 */24 0/24 
agent 2: 10/24 */24 */24 2/24 
agent 3: 2/24 */24 */24 11/24 
agent 4: 2/24 */24 */24 11/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 */24 
agent 2: 4/24 10/24 8/24 2/24 
agent 3: */24 2/24 8/24 10/24 
agent 4: */24 2/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 2/24 
agent 2: */24 16/24 0/24 6/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 16/24 0/24 6/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > c > d > a
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: */24 0/24 16/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 0/24 12/24 4/24 8/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 2/24 0/24 16/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 6/24 14/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 6/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 */24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 6/24 14/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 6/24 14/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 2/24 5/24 9/24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 6/24 14/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 */24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 10/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 */24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 6/24 4/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 4/24 */24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 10/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 */24 */24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 10/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 2/24 10/24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 10/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: */24 */24 */24 */24 
agent 3: 2/24 10/24 */24 */24 
agent 4: */24 10/24 2/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: */24 2/24 */24 */24 
agent 3: 2/24 10/24 */24 */24 
agent 4: */24 10/24 2/24 */24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: */24 2/24 4/24 8/24 
agent 3: 2/24 10/24 4/24 8/24 
agent 4: 2/24 10/24 4/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: */24 2/24 4/24 8/24 
agent 3: 2/24 10/24 4/24 8/24 
agent 4: 2/24 10/24 4/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: 10/24 2/24 4/24 8/24 
agent 3: 2/24 10/24 4/24 8/24 
agent 4: 2/24 10/24 4/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: 10/24 0/24 4/24 */24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: */24 11/24 4/24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 2/24 */24 */24 
agent 3: 2/24 10/24 */24 */24 
agent 4: */24 10/24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: 10/24 0/24 4/24 */24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: */24 11/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: 10/24 0/24 4/24 10/24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: */24 11/24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: 10/24 0/24 4/24 10/24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: 2/24 11/24 4/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: 10/24 0/24 4/24 10/24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: */24 11/24 4/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: 10/24 0/24 4/24 10/24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: */24 11/24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 12/24 0/24 
agent 2: 10/24 0/24 4/24 10/24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: 2/24 11/24 4/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 0/24 12/24 2/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: */24 11/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 0/24 12/24 2/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: */24 11/24 1/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 0/24 12/24 2/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 2/24 11/24 1/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 0/24 12/24 */24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: */24 11/24 1/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 0/24 12/24 */24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: */24 11/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 0/24 12/24 2/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: */24 11/24 1/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 0/24 12/24 2/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 2/24 11/24 1/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 2/24 */24 2/24 
agent 3: 2/24 10/24 */24 */24 
agent 4: */24 10/24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 2/24 */24 2/24 
agent 3: 2/24 10/24 */24 */24 
agent 4: */24 10/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 2/24 10/24 2/24 
agent 3: 2/24 10/24 */24 */24 
agent 4: */24 10/24 2/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 10/24 2/24 
agent 2: 10/24 2/24 10/24 2/24 
agent 3: 2/24 10/24 2/24 10/24 
agent 4: 2/24 10/24 2/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 0/24 */24 */24 
agent 2: 4/24 8/24 10/24 2/24 
agent 3: 4/24 8/24 2/24 10/24 
agent 4: */24 8/24 2/24 */24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > a > d > c
agent 4: b > a > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 0/24 */24 */24 
agent 2: 4/24 8/24 10/24 2/24 
agent 3: 4/24 8/24 2/24 10/24 
agent 4: */24 8/24 2/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 0/24 10/24 */24 
agent 2: 4/24 8/24 10/24 2/24 
agent 3: 4/24 8/24 2/24 10/24 
agent 4: 4/24 8/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 0/24 10/24 2/24 
agent 2: 4/24 8/24 10/24 2/24 
agent 3: 4/24 8/24 2/24 10/24 
agent 4: 4/24 8/24 2/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > b > d > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 6/24 10/24 2/24 
agent 2: 6/24 6/24 10/24 2/24 
agent 3: 6/24 6/24 */24 */24 
agent 4: 6/24 6/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 2/24 
agent 2: 4/24 10/24 8/24 2/24 
agent 3: */24 2/24 8/24 10/24 
agent 4: */24 2/24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > b > d > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 6/24 10/24 2/24 
agent 2: 6/24 6/24 10/24 2/24 
agent 3: 6/24 6/24 */24 */24 
agent 4: 6/24 6/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 6/24 10/24 2/24 
agent 2: 6/24 6/24 10/24 2/24 
agent 3: 6/24 6/24 2/24 10/24 
agent 4: 6/24 6/24 2/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > c > d > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 */24 0/24 2/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 6/24 */24 8/24 */24 
agent 4: 6/24 1/24 8/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: b > a > d > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 4/24 10/24 2/24 
agent 2: 8/24 4/24 10/24 2/24 
agent 3: 8/24 4/24 2/24 10/24 
agent 4: 0/24 12/24 2/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: a > c > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 0/24 2/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 6/24 */24 8/24 */24 
agent 4: 6/24 1/24 8/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 16/24 0/24 2/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 6/24 */24 8/24 */24 
agent 4: 6/24 1/24 8/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 16/24 0/24 2/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 6/24 1/24 8/24 9/24 
agent 4: 6/24 1/24 8/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 0/24 2/24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 16/24 */24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 6/24 */24 */24 8/24 
agent 4: 6/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 0/24 2/24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 0/24 2/24 
agent 2: 2/24 8/24 8/24 6/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: */24 8/24 6/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: */24 8/24 6/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 8/24 6/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 16/24 2/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 6/24 */24 */24 8/24 
agent 4: 6/24 */24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: */24 */24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 16/24 2/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 6/24 */24 */24 8/24 
agent 4: 6/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 16/24 2/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 6/24 1/24 9/24 8/24 
agent 4: 6/24 1/24 9/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 1/24 */24 5/24 
agent 4: 0/24 1/24 9/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 1/24 */24 5/24 
agent 4: 0/24 1/24 9/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 8/24 3/24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 1/24 9/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 */24 0/24 
agent 2: 10/24 */24 */24 2/24 
agent 3: 2/24 1/24 10/24 11/24 
agent 4: 2/24 */24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 */24 0/24 
agent 2: 10/24 */24 */24 2/24 
agent 3: 2/24 1/24 10/24 11/24 
agent 4: 2/24 */24 */24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 */24 */24 2/24 
agent 3: 2/24 1/24 10/24 11/24 
agent 4: 2/24 */24 */24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: */24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 */24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 2/24 8/24 */24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: */24 */24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 */24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 */24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: */24 */24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 2/24 6/24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 2/24 6/24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 */24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 6/24 8/24 */24 
agent 3: 0/24 */24 8/24 10/24 
agent 4: 0/24 6/24 8/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > c > a > b
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 8/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 2/24 8/24 */24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 6/24 */24 8/24 
agent 3: 0/24 6/24 10/24 8/24 
agent 4: 0/24 */24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 2/24 8/24 */24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 8/24 6/24 8/24 
agent 3: 0/24 8/24 8/24 8/24 
agent 4: 0/24 8/24 8/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 4/24 */24 2/24 8/24 
agent 3: 4/24 2/24 10/24 8/24 
agent 4: */24 2/24 */24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 */24 6/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 2/24 4/24 0/24 
agent 2: 6/24 */24 6/24 8/24 
agent 3: 0/24 9/24 7/24 8/24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 4/24 */24 2/24 8/24 
agent 3: 4/24 2/24 10/24 8/24 
agent 4: */24 2/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 2/24 0/24 
agent 2: 4/24 10/24 2/24 8/24 
agent 3: 4/24 2/24 10/24 8/24 
agent 4: */24 2/24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: */24 10/24 2/24 0/24 
agent 2: 4/24 10/24 2/24 8/24 
agent 3: 4/24 2/24 10/24 8/24 
agent 4: */24 2/24 10/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 10/24 2/24 0/24 
agent 2: 4/24 10/24 2/24 8/24 
agent 3: 4/24 2/24 10/24 8/24 
agent 4: 4/24 2/24 10/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 4/24 0/24 
agent 2: 4/24 8/24 4/24 8/24 
agent 3: */24 */24 4/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 10/24 0/24 2/24 
agent 2: 4/24 10/24 8/24 2/24 
agent 3: */24 2/24 8/24 10/24 
agent 4: */24 2/24 8/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 10/24 2/24 2/24 
agent 3: 2/24 1/24 10/24 11/24 
agent 4: 2/24 */24 */24 11/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 4/24 0/24 
agent 2: 4/24 8/24 4/24 8/24 
agent 3: 4/24 */24 4/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > a > b > c
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 4/24 0/24 
agent 2: 4/24 8/24 4/24 8/24 
agent 3: 4/24 */24 4/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 8/24 4/24 0/24 
agent 2: 4/24 8/24 4/24 8/24 
agent 3: 4/24 8/24 4/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 8/24 0/24 4/24 
agent 2: */24 8/24 8/24 4/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > b > d
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 8/24 0/24 4/24 
agent 2: */24 8/24 8/24 4/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 8/24 0/24 4/24 
agent 2: 4/24 8/24 8/24 4/24 
agent 3: */24 */24 8/24 4/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 8/24 0/24 4/24 
agent 2: 4/24 8/24 8/24 4/24 
agent 3: 4/24 8/24 8/24 4/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 10/24 0/24 2/24 
agent 2: 4/24 10/24 8/24 2/24 
agent 3: 4/24 2/24 8/24 10/24 
agent 4: */24 2/24 8/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 10/24 0/24 2/24 
agent 2: 4/24 10/24 8/24 2/24 
agent 3: 4/24 2/24 8/24 10/24 
agent 4: */24 2/24 8/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 10/24 0/24 2/24 
agent 2: 4/24 10/24 8/24 2/24 
agent 3: 4/24 2/24 8/24 10/24 
agent 4: 4/24 2/24 8/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 10/24 2/24 2/24 
agent 3: 2/24 1/24 10/24 11/24 
agent 4: 2/24 */24 */24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 10/24 2/24 2/24 
agent 3: 2/24 1/24 10/24 11/24 
agent 4: 2/24 1/24 10/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 */24 6/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 6/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 16/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 */24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 0/24 8/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 22/24 2/24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 0/24 
agent 3: 0/24 3/24 9/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 0/24 
agent 3: 0/24 3/24 9/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 3/24 9/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 3/24 9/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 10/24 11/24 
agent 4: 0/24 */24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 10/24 11/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 3/24 10/24 11/24 
agent 4: 0/24 3/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 3/24 9/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 23/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 6/24 0/24 
agent 3: 0/24 3/24 9/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 */24 0/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 */24 0/24 
agent 3: 0/24 */24 11/24 12/24 
agent 4: 0/24 1/24 */24 12/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 */24 0/24 
agent 3: 0/24 */24 11/24 12/24 
agent 4: 0/24 1/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 2/24 0/24 
agent 3: 0/24 1/24 11/24 12/24 
agent 4: 0/24 1/24 11/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 3/24 11/24 10/24 
agent 4: 0/24 3/24 */24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 2/24 */24 
agent 3: 0/24 3/24 11/24 10/24 
agent 4: 0/24 3/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 */24 
agent 3: 0/24 3/24 11/24 10/24 
agent 4: 0/24 3/24 11/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 11/24 10/24 
agent 4: 0/24 3/24 11/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 0/24 */24 8/24 10/24 
agent 4: 0/24 6/24 8/24 10/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: 0/24 */24 12/24 9/24 
agent 4: 0/24 3/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 0/24 */24 8/24 10/24 
agent 4: 0/24 6/24 8/24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 6/24 8/24 4/24 
agent 3: 0/24 6/24 8/24 10/24 
agent 4: 0/24 6/24 8/24 10/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 */24 
agent 3: 0/24 */24 12/24 9/24 
agent 4: 0/24 3/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 3/24 12/24 9/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 9/24 
agent 4: 0/24 3/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 */24 12/24 9/24 
agent 4: 0/24 3/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 0/24 6/24 
agent 3: 0/24 3/24 12/24 9/24 
agent 4: 0/24 3/24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 */24 0/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 9/24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 */24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 */24 9/24 0/24 
agent 3: 0/24 9/24 */24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 */24 
agent 2: 12/24 */24 0/24 9/24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 */24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 */24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 0/24 */24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 23/24 0/24 1/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 0/24 9/24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 */24 0/24 9/24 
agent 3: 0/24 */24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 0/24 */24 9/24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 3/24 
agent 2: 12/24 0/24 */24 9/24 
agent 3: 0/24 11/24 */24 2/24 
agent 4: 0/24 11/24 */24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 */24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 */24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 */24 10/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 */24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 4/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 0/24 4/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 23/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 23/24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 */24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 */24 */24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 14/24 2/24 
agent 3: 8/24 0/24 1/24 15/24 
agent 4: 0/24 18/24 2/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 */24 10/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 3/24 
agent 2: 8/24 12/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 2/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 */24 10/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 0/24 3/24 9/24 
agent 3: 0/24 11/24 11/24 2/24 
agent 4: 0/24 11/24 3/24 10/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 9/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 0/24 9/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 */24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 0/24 3/24 
agent 2: 0/24 21/24 0/24 3/24 
agent 3: 4/24 2/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 */24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 */24 */24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 */24 */24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 9/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 */24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 9/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 1/24 */24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 */24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 9/24 3/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 21/24 3/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 5/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 5/24 0/24 
agent 4: 0/24 */24 3/24 20/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 */24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 */24 0/24 
agent 4: 0/24 */24 3/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 5/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 */24 5/24 0/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 4/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 */24 */24 
agent 4: 0/24 1/24 */24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 */24 5/24 0/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 5/24 0/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 */24 4/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 5/24 0/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 */24 0/24 
agent 2: 8/24 11/24 4/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 2/24 */24 17/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 0/24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 */24 0/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 */24 0/24 
agent 2: 8/24 11/24 4/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 2/24 */24 17/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 4/24 1/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 2/24 5/24 17/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 */24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 1/24 */24 19/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 0/24 
agent 2: 8/24 5/24 4/24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 19/24 0/24 5/24 0/24 
agent 2: 4/24 12/24 6/24 2/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 5/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 */24 0/24 
agent 2: 11/24 0/24 5/24 8/24 
agent 3: 2/24 11/24 4/24 */24 
agent 4: 0/24 11/24 */24 9/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 2/24 */24 0/24 
agent 2: 11/24 0/24 5/24 8/24 
agent 3: 2/24 11/24 4/24 */24 
agent 4: 0/24 11/24 */24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 2/24 11/24 0/24 
agent 2: 11/24 0/24 5/24 8/24 
agent 3: 2/24 11/24 4/24 7/24 
agent 4: 0/24 11/24 4/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 */24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > c > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 11/24 0/24 
agent 2: 8/24 5/24 4/24 7/24 
agent 3: 8/24 0/24 4/24 12/24 
agent 4: 0/24 14/24 5/24 5/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > d > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 0/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 6/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 */24 18/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 10/24 */24 
agent 4: 0/24 0/24 6/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 */24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 6/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 0/24 6/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 6/24 */24 
agent 4: 0/24 0/24 6/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 10/24 6/24 0/24 
agent 2: 8/24 10/24 6/24 0/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 6/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 6/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 */24 0/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 11/24 5/24 0/24 
agent 2: 8/24 11/24 5/24 0/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 2/24 4/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 5/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 5/24 0/24 
agent 4: 0/24 */24 3/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 5/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 */24 
agent 4: 0/24 1/24 4/24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 */24 
agent 4: 0/24 1/24 4/24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 */24 1/24 
agent 4: 0/24 1/24 4/24 19/24 

Assume agent 4 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 */24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 */24 1/24 
agent 4: 0/24 1/24 4/24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 4/24 1/24 
agent 4: 0/24 1/24 4/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 1/24 
agent 4: 0/24 0/24 */24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 3/24 11/24 
agent 4: 0/24 2/24 11/24 11/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 5/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 */24 */24 
agent 3: 0/24 10/24 3/24 11/24 
agent 4: 0/24 2/24 11/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 2/24 
agent 3: 0/24 10/24 3/24 11/24 
agent 4: 0/24 2/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: 6/24 3/24 7/24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 7/24 3/24 2/24 
agent 3: 0/24 8/24 */24 11/24 
agent 4: 0/24 0/24 */24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 4/24 */24 0/24 
agent 2: 6/24 3/24 7/24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 3/24 7/24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 18/24 4/24 2/24 0/24 
agent 2: 6/24 3/24 7/24 8/24 
agent 3: 0/24 15/24 1/24 8/24 
agent 4: 0/24 2/24 14/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 */24 0/24 
agent 2: 12/24 7/24 3/24 2/24 
agent 3: 0/24 8/24 */24 11/24 
agent 4: 0/24 0/24 */24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 2/24 
agent 3: 0/24 8/24 5/24 11/24 
agent 4: 0/24 0/24 13/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: d > a > c > b
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 23/24 1/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 0/24 12/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 2/24 6/24 4/24 
agent 3: 0/24 18/24 5/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 */24 
agent 3: 0/24 8/24 5/24 11/24 
agent 4: 0/24 0/24 13/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 */24 
agent 3: 0/24 8/24 5/24 11/24 
agent 4: 0/24 0/24 13/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 7/24 3/24 2/24 
agent 3: 0/24 8/24 5/24 11/24 
agent 4: 0/24 0/24 13/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 2/24 
agent 3: 0/24 10/24 */24 11/24 
agent 4: 0/24 2/24 11/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > b > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 2/24 
agent 3: 0/24 10/24 */24 11/24 
agent 4: 0/24 2/24 11/24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 9/24 3/24 0/24 
agent 2: 12/24 3/24 7/24 2/24 
agent 3: 0/24 10/24 3/24 11/24 
agent 4: 0/24 2/24 11/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 5/24 19/24 

Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 */24 
agent 4: 0/24 0/24 5/24 19/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 5/24 19/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > d > a > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 */24 0/24 
agent 4: 0/24 1/24 3/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 5/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 8/24 4/24 
agent 3: 0/24 19/24 5/24 0/24 
agent 4: 0/24 1/24 3/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 */24 2/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 */24 
agent 4: 0/24 1/24 0/24 23/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 */24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 */24 
agent 4: 0/24 1/24 0/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 12/24 0/24 
agent 3: 8/24 3/24 12/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 */24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 */24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > c > b
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: */24 0/24 1/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 0/24 1/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 23/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 23/24 */24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 1/24 3/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > b > c > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 23/24 1/24 0/24 0/24 
agent 2: 1/24 3/24 20/24 0/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 15/24 0/24 */24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 0/24 1/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 2/24 0/24 14/24 
agent 4: 0/24 4/24 18/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 5/24 */24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > d > b > c
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 */24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 10/24 1/24 
agent 2: 8/24 5/24 3/24 8/24 
agent 3: 8/24 0/24 2/24 14/24 
agent 4: 0/24 14/24 9/24 1/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > d > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 0/24 12/24 1/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 11/24 1/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 6/24 0/24 
agent 3: 0/24 3/24 9/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 3/24 9/24 12/24 
agent 4: 0/24 3/24 9/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 6/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 6/24 2/24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 */24 */24 11/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 6/24 */24 0/24 
agent 3: 0/24 6/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 */24 6/24 2/24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 */24 */24 11/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 */24 */24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 4/24 6/24 2/24 
agent 3: 0/24 7/24 6/24 11/24 
agent 4: 0/24 7/24 6/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 2/24 4/24 0/24 
agent 2: 6/24 4/24 6/24 8/24 
agent 3: 0/24 9/24 7/24 8/24 
agent 4: 0/24 */24 */24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 2/24 4/24 0/24 
agent 2: 6/24 4/24 6/24 8/24 
agent 3: 0/24 9/24 7/24 8/24 
agent 4: 0/24 */24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 2/24 4/24 0/24 
agent 2: 6/24 4/24 6/24 8/24 
agent 3: 0/24 9/24 7/24 8/24 
agent 4: 0/24 9/24 7/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 6/24 */24 0/24 
agent 3: 0/24 6/24 */24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 22/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 22/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 12/24 2/24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 22/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 12/24 2/24 0/24 
agent 3: 2/24 0/24 */24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 12/24 2/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 12/24 2/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 2/24 0/24 10/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 0/24 6/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 22/24 */24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 */24 
agent 2: 0/24 22/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 */24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 */24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 22/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 22/24 0/24 */24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 22/24 0/24 2/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 */24 
agent 2: 0/24 22/24 0/24 2/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 */24 */24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 0/24 10/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 18/24 0/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 0/24 10/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 18/24 0/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 10/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 18/24 0/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 */24 
agent 4: 0/24 11/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 */24 */24 
agent 4: 0/24 0/24 21/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 */24 12/24 */24 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 */24 */24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 0/24 8/24 
agent 3: 8/24 8/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 4/24 12/24 0/24 8/24 
agent 3: 4/24 12/24 0/24 8/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 0/24 10/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 18/24 0/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 0/24 10/24 */24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 18/24 0/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 0/24 10/24 6/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 18/24 0/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 */24 
agent 4: 0/24 11/24 */24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 */24 
agent 4: 0/24 11/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 11/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 */24 12/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > d > a
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 0/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 9/24 12/24 3/24 
agent 4: 0/24 9/24 12/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 22/24 0/24 2/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: c > a > d > b
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 */24 0/24 6/24 
agent 2: 0/24 22/24 0/24 2/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 0/24 6/24 
agent 2: 0/24 22/24 0/24 2/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 4/24 0/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 22/24 2/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 4/24 0/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 9/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 2/24 16/24 6/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 */24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 */24 22/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 4/24 8/24 0/24 
agent 3: 0/24 16/24 6/24 2/24 
agent 4: 0/24 0/24 2/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 2/24 10/24 0/24 
agent 3: 0/24 10/24 */24 12/24 
agent 4: 0/24 */24 2/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 2/24 10/24 0/24 
agent 3: 0/24 10/24 */24 12/24 
agent 4: 0/24 */24 2/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 2/24 10/24 0/24 
agent 3: 0/24 10/24 2/24 12/24 
agent 4: 0/24 10/24 2/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: d > b > c > a
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 */24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 8/24 0/24 
agent 3: 0/24 4/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 6/24 */24 8/24 
agent 3: 0/24 6/24 10/24 8/24 
agent 4: 0/24 */24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 0/24 6/24 10/24 8/24 
agent 4: 0/24 */24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 6/24 4/24 8/24 
agent 3: 0/24 6/24 10/24 8/24 
agent 4: 0/24 6/24 10/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 3/24 10/24 11/24 
agent 4: 0/24 3/24 */24 */24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 */24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 */24 
agent 3: 0/24 3/24 10/24 11/24 
agent 4: 0/24 3/24 */24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 10/24 11/24 
agent 4: 0/24 3/24 */24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 10/24 11/24 
agent 4: 0/24 3/24 10/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 2/24 
agent 3: 0/24 1/24 12/24 11/24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 2/24 
agent 3: 0/24 1/24 12/24 11/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 0/24 2/24 
agent 3: 0/24 1/24 12/24 11/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 10/24 0/24 2/24 
agent 3: 0/24 1/24 12/24 11/24 
agent 4: 0/24 1/24 12/24 11/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 10/24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > b > c
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 10/24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 6/24 10/24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > a > c
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > b > a
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > b > c > a
agent 3: d > c > b > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 */24 12/24 */24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 16/24 0/24 8/24 
agent 3: 0/24 4/24 12/24 8/24 
agent 4: 0/24 4/24 12/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 22/24 0/24 */24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 22/24 0/24 2/24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 6/24 4/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 6/24 4/24 14/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 2/24 0/24 
agent 2: 8/24 2/24 9/24 5/24 
agent 3: 8/24 2/24 9/24 5/24 
agent 4: 0/24 6/24 4/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 */24 0/24 
agent 2: 8/24 2/24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 15/24 */24 0/24 
agent 2: 8/24 2/24 */24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: b > d > c > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 23/24 0/24 1/24 0/24 
agent 2: 0/24 12/24 12/24 0/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 1/24 0/24 3/24 20/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 */24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > d > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 */24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > a > b > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 15/24 1/24 0/24 
agent 2: 8/24 2/24 14/24 0/24 
agent 3: 8/24 3/24 7/24 6/24 
agent 4: 0/24 4/24 2/24 18/24 

Assume agent 3 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > d > a
agent 3: d > a > b > c
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 6/24 2/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > c > b
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 12/24 2/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: */24 0/24 */24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 12/24 2/24 0/24 
agent 2: 10/24 12/24 2/24 0/24 
agent 3: 2/24 0/24 10/24 12/24 
agent 4: 2/24 0/24 10/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: 10/24 6/24 6/24 2/24 
agent 3: */24 5/24 6/24 */24 
agent 4: 2/24 5/24 6/24 11/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: 10/24 6/24 6/24 2/24 
agent 3: 2/24 5/24 6/24 11/24 
agent 4: 2/24 5/24 6/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: */24 0/24 */24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > d > a > c
agent 3: b > d > c > a
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 4/24 0/24 
agent 2: 2/24 12/24 6/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 2/24 0/24 6/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: d > a > c > b
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 6/24 0/24 10/24 8/24 
agent 3: 1/24 */24 4/24 8/24 
agent 4: */24 11/24 */24 8/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 6/24 0/24 10/24 8/24 
agent 3: 1/24 11/24 4/24 8/24 
agent 4: */24 11/24 */24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 6/24 0/24 10/24 8/24 
agent 3: 1/24 11/24 4/24 8/24 
agent 4: 1/24 11/24 4/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 0/24 */24 6/24 
agent 2: 6/24 8/24 */24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 1/24 8/24 9/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 6/24 8/24 4/24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 1/24 8/24 9/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 0/24 2/24 6/24 
agent 2: 6/24 8/24 4/24 6/24 
agent 3: 1/24 8/24 9/24 6/24 
agent 4: 1/24 8/24 9/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > b > d
The allocation takes the form
         a   b   c   d
agent 1: 6/24 10/24 2/24 6/24 
agent 2: 6/24 10/24 */24 6/24 
agent 3: 6/24 */24 */24 6/24 
agent 4: 6/24 2/24 10/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > c > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 10/24 2/24 6/24 
agent 2: 6/24 10/24 */24 6/24 
agent 3: 6/24 */24 */24 6/24 
agent 4: 6/24 2/24 10/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 10/24 2/24 6/24 
agent 2: 6/24 10/24 2/24 6/24 
agent 3: 6/24 */24 */24 6/24 
agent 4: 6/24 2/24 10/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 10/24 2/24 6/24 
agent 2: 6/24 10/24 2/24 6/24 
agent 3: 6/24 2/24 10/24 6/24 
agent 4: 6/24 2/24 10/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 9/24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 */24 */24 6/24 
agent 4: 0/24 */24 */24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 10/24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 10/24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > c > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 */24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: 6/24 */24 4/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 18/24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: 6/24 2/24 4/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 18/24 0/24 0/24 
agent 2: 6/24 */24 12/24 4/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 18/24 0/24 0/24 
agent 2: 6/24 */24 12/24 4/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 18/24 0/24 0/24 
agent 2: 6/24 2/24 12/24 4/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 10/24 8/24 0/24 
agent 2: 6/24 10/24 */24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 2/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 10/24 8/24 0/24 
agent 2: 6/24 10/24 8/24 0/24 
agent 3: 6/24 */24 */24 12/24 
agent 4: 6/24 2/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 10/24 8/24 0/24 
agent 2: 6/24 10/24 8/24 0/24 
agent 3: 6/24 2/24 4/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > c > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 16/24 8/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 */24 */24 12/24 
agent 4: 0/24 16/24 8/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 1/24 3/24 12/24 
agent 4: 0/24 16/24 8/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 */24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 6/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 2/24 6/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 */24 */24 12/24 
agent 4: 4/24 2/24 6/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: 6/24 8/24 4/24 6/24 
agent 4: 6/24 0/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: 6/24 8/24 4/24 6/24 
agent 4: 6/24 0/24 2/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: */24 8/24 8/24 */24 
agent 3: 6/24 8/24 8/24 2/24 
agent 4: 6/24 0/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 9/24 1/24 
agent 2: 6/24 8/24 9/24 1/24 
agent 3: 6/24 8/24 4/24 6/24 
agent 4: 6/24 0/24 2/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: */24 8/24 8/24 */24 
agent 3: 6/24 8/24 8/24 2/24 
agent 4: 6/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: */24 8/24 8/24 */24 
agent 3: 6/24 8/24 8/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: 6/24 8/24 8/24 2/24 
agent 3: 6/24 8/24 8/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > b > c
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 16/24 2/24 
agent 4: 6/24 0/24 0/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: */24 8/24 8/24 */24 
agent 3: 6/24 8/24 8/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 16/24 2/24 
agent 4: 6/24 0/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 16/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: 6/24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 */24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 16/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: 6/24 8/24 */24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: 6/24 8/24 7/24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 7/24 3/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 7/24 3/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 7/24 3/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 3/24 7/24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 3/24 7/24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 3/24 7/24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 3/24 7/24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 */24 3/24 7/24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 3/24 7/24 
agent 3: 0/24 10/24 12/24 2/24 
agent 4: 0/24 10/24 2/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 */24 3/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 */24 3/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 7/24 3/24 
agent 2: 12/24 2/24 7/24 3/24 
agent 3: 0/24 10/24 10/24 4/24 
agent 4: 0/24 10/24 0/24 14/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 18/24 0/24 */24 2/24 
agent 2: 6/24 8/24 7/24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > a > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 0/24 */24 2/24 
agent 2: 6/24 8/24 7/24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 0/24 4/24 2/24 
agent 2: 6/24 8/24 7/24 3/24 
agent 3: 0/24 8/24 13/24 3/24 
agent 4: 0/24 8/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 16/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: */24 8/24 8/24 */24 
agent 3: 6/24 8/24 8/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 8/24 2/24 
agent 2: 6/24 8/24 8/24 2/24 
agent 3: 6/24 8/24 8/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 */24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 7/24 6/24 
agent 4: 0/24 11/24 7/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 */24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 4/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 9/24 6/24 
agent 3: 8/24 1/24 9/24 6/24 
agent 4: 0/24 16/24 2/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 0/24 
agent 2: 8/24 */24 4/24 11/24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 */24 0/24 
agent 2: 8/24 */24 4/24 11/24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 4/24 11/24 
agent 3: 8/24 1/24 4/24 11/24 
agent 4: 0/24 16/24 6/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 6/24 6/24 
agent 3: 0/24 11/24 4/24 9/24 
agent 4: 0/24 11/24 4/24 9/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > c > a
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 */24 
agent 4: 0/24 1/24 18/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 */24 18/24 5/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 */24 18/24 5/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 0/24 5/24 
agent 4: 0/24 1/24 18/24 5/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > d > b
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 10/24 5/24 
agent 3: 8/24 0/24 10/24 6/24 
agent 4: 0/24 17/24 0/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 */24 14/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 6/24 14/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 2/24 5/24 9/24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 6/24 14/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 6/24 14/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 */24 5/24 */24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 6/24 14/24 4/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 2/24 5/24 9/24 
agent 3: 8/24 2/24 5/24 9/24 
agent 4: 0/24 6/24 14/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: b > c > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 10/24 */24 */24 
agent 4: 0/24 10/24 8/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 10/24 8/24 6/24 
agent 4: 0/24 10/24 8/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > c > d > b
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 */24 12/24 4/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 6/24 2/24 12/24 4/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 6/24 18/24 0/24 0/24 
agent 2: 6/24 2/24 12/24 4/24 
agent 3: 6/24 2/24 12/24 4/24 
agent 4: 6/24 2/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 18/24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: 6/24 2/24 4/24 12/24 
agent 4: */24 2/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 18/24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: 6/24 2/24 4/24 12/24 
agent 4: */24 2/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 18/24 0/24 0/24 
agent 2: 6/24 2/24 16/24 0/24 
agent 3: 6/24 2/24 4/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 17/24 1/24 0/24 
agent 2: 6/24 1/24 */24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 17/24 1/24 0/24 
agent 2: 6/24 1/24 17/24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 17/24 1/24 0/24 
agent 2: 6/24 */24 17/24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 17/24 1/24 0/24 
agent 2: 6/24 */24 17/24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 17/24 1/24 0/24 
agent 2: 6/24 1/24 17/24 0/24 
agent 3: 6/24 4/24 2/24 12/24 
agent 4: 6/24 2/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 1/24 23/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > d > a
agent 4: d > a > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 */24 */24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 1/24 23/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 */24 23/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 */24 23/24 0/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 1/24 23/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > c > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 4/24 0/24 12/24 
agent 3: 8/24 3/24 1/24 12/24 
agent 4: 0/24 1/24 23/24 0/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 */24 12/24 */24 
agent 4: 0/24 3/24 12/24 9/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 11/24 11/24 0/24 2/24 
agent 2: 11/24 3/24 0/24 10/24 
agent 3: 2/24 7/24 12/24 3/24 
agent 4: 0/24 3/24 12/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 7/24 */24 3/24 
agent 3: 8/24 */24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 7/24 */24 3/24 
agent 3: 8/24 */24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 14/24 0/24 2/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 1/24 0/24 15/24 
agent 4: 0/24 2/24 18/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: a > d > b > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 16/24 0/24 0/24 
agent 2: 8/24 3/24 6/24 7/24 
agent 3: 8/24 3/24 0/24 13/24 
agent 4: 0/24 2/24 18/24 4/24 

Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > a > b
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 */24 
agent 4: 0/24 1/24 0/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 */24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 1/24 0/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 */24 18/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 18/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > a > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > b > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 8/24 13/24 0/24 3/24 
agent 2: 8/24 7/24 6/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 4/24 18/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 7/24 0/24 */24 
agent 2: 11/24 7/24 0/24 6/24 
agent 3: */24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 7/24 0/24 6/24 
agent 2: 11/24 7/24 0/24 6/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 6/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 2/24 4/24 12/24 6/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 9/24 0/24 4/24 
agent 2: 11/24 9/24 0/24 4/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 6/24 12/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: c > b > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 */24 
agent 4: 2/24 4/24 12/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 17/24 1/24 0/24 6/24 
agent 2: 1/24 19/24 0/24 4/24 
agent 3: 4/24 0/24 12/24 8/24 
agent 4: 2/24 4/24 12/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > a > c > d
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 19/24 1/24 4/24 
agent 4: 0/24 1/24 17/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 */24 */24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 10/24 1/24 5/24 
agent 2: 8/24 10/24 1/24 5/24 
agent 3: 8/24 4/24 6/24 6/24 
agent 4: 0/24 0/24 16/24 8/24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > d > b
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: 11/24 12/24 0/24 1/24 
agent 3: */24 0/24 12/24 10/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 12/24 0/24 1/24 
agent 2: 11/24 12/24 0/24 1/24 
agent 3: 2/24 0/24 12/24 10/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > a > b > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 11/24 10/24 0/24 3/24 
agent 2: 11/24 10/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: 10/24 8/24 6/24 0/24 
agent 3: */24 4/24 6/24 12/24 
agent 4: 2/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 10/24 8/24 6/24 0/24 
agent 2: 10/24 8/24 6/24 0/24 
agent 3: 2/24 4/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > a > c > d
agent 4: b > a > c > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: */24 11/24 8/24 */24 
agent 4: 2/24 11/24 8/24 3/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 10/24 2/24 8/24 4/24 
agent 2: 10/24 0/24 0/24 14/24 
agent 3: 2/24 11/24 8/24 3/24 
agent 4: 2/24 11/24 8/24 3/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 */24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 */24 */24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: b > a > d > c
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 14/24 0/24 */24 0/24 
agent 2: 5/24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 10/24 0/24 
agent 2: 5/24 8/24 5/24 6/24 
agent 3: */24 8/24 */24 6/24 
agent 4: 0/24 8/24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 14/24 0/24 10/24 0/24 
agent 2: 5/24 8/24 5/24 6/24 
agent 3: 5/24 8/24 5/24 6/24 
agent 4: 0/24 8/24 4/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > c > b
The allocation takes the form
         a   b   c   d
agent 1: 6/24 8/24 10/24 0/24 
agent 2: 6/24 8/24 */24 5/24 
agent 3: 6/24 */24 5/24 5/24 
agent 4: 6/24 0/24 4/24 14/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: a > d > c > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 8/24 10/24 0/24 
agent 2: 6/24 8/24 */24 5/24 
agent 3: 6/24 */24 5/24 5/24 
agent 4: 6/24 0/24 4/24 14/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 8/24 10/24 0/24 
agent 2: 6/24 8/24 5/24 5/24 
agent 3: 6/24 8/24 5/24 5/24 
agent 4: 6/24 0/24 4/24 14/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: a > d > b > c
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 */24 */24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 1/24 3/24 12/24 
agent 4: 0/24 16/24 8/24 0/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 10/24 0/24 
agent 2: 8/24 1/24 3/24 12/24 
agent 3: 8/24 1/24 3/24 12/24 
agent 4: 0/24 16/24 8/24 0/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 11/24 1/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 */24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > a > c
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 11/24 1/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 10/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 11/24 1/24 12/24 
agent 4: 0/24 11/24 1/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > c > a > d
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 2/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 */24 3/24 
agent 4: 0/24 11/24 10/24 3/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 2/24 4/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 11/24 10/24 3/24 
agent 4: 0/24 11/24 10/24 3/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > a > c
The allocation takes the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > c > b > d
agent 4: b > d > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 */24 */24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 4/24 6/24 
agent 2: 8/24 1/24 10/24 5/24 
agent 3: 8/24 1/24 10/24 5/24 
agent 4: 0/24 16/24 0/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: b > d > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 1/24 */24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 9/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 5/24 8/24 3/24 
agent 2: 8/24 5/24 1/24 10/24 
agent 3: 8/24 0/24 14/24 2/24 
agent 4: 0/24 14/24 1/24 9/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: b > d > c > a
The allocation takes the form
         a   b   c   d
agent 1: 11/24 2/24 10/24 1/24 
agent 2: 11/24 0/24 12/24 1/24 
agent 3: 2/24 11/24 1/24 10/24 
agent 4: 0/24 11/24 1/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 17/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 17/24 7/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 */24 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 */24 */24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 */24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > c > d > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 4/24 8/24 
agent 4: 0/24 0/24 17/24 7/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > b > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 1/24 */24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 18/24 1/24 5/24 0/24 
agent 2: 0/24 17/24 7/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 2/24 4/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > b > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 11/24 8/24 5/24 0/24 
agent 2: 11/24 0/24 13/24 0/24 
agent 3: 2/24 7/24 3/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: a > d > c > b
agent 4: b > c > a > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 6/24 7/24 3/24 
agent 2: 8/24 0/24 13/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 18/24 4/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: b > c > d > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 */24 
agent 4: 0/24 0/24 17/24 7/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 4/24 8/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 17/24 7/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 9/24 0/24 3/24 
agent 2: 12/24 3/24 2/24 7/24 
agent 3: 0/24 10/24 11/24 3/24 
agent 4: 0/24 2/24 11/24 11/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: a > d > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 16/24 */24 
agent 4: 6/24 0/24 0/24 18/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 6/24 12/24 4/24 2/24 
agent 2: 6/24 12/24 4/24 2/24 
agent 3: 6/24 0/24 16/24 2/24 
agent 4: 6/24 0/24 0/24 18/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > d > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > d > b > c
agent 4: c > a > d > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 */24 */24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 1/24 3/24 
agent 2: 8/24 12/24 1/24 3/24 
agent 3: 8/24 0/24 0/24 16/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: b > d > a > c
agent 4: c > b > d > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 0/24 1/24 3/24 
agent 2: 4/24 12/24 2/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > d > a
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 */24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 3/24 21/24 0/24 
agent 4: 0/24 3/24 1/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > c > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: */24 12/24 */24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 16/24 8/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 0/24 0/24 4/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 2/24 12/24 4/24 6/24 
agent 4: 0/24 0/24 16/24 8/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 2/24 */24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 */24 6/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 2/24 6/24 0/24 
agent 2: 0/24 18/24 6/24 0/24 
agent 3: 4/24 2/24 6/24 12/24 
agent 4: 4/24 2/24 6/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 6/24 */24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: d > b > a > c
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 */24 6/24 12/24 
agent 4: 0/24 6/24 */24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 6/24 6/24 0/24 
agent 2: 12/24 6/24 6/24 0/24 
agent 3: 0/24 6/24 6/24 12/24 
agent 4: 0/24 6/24 6/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > d > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > d > c > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 2/24 4/24 6/24 
agent 3: 0/24 18/24 0/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > c > a > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: b > d > c > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 4/24 2/24 6/24 
agent 3: 0/24 16/24 0/24 8/24 
agent 4: 0/24 0/24 20/24 4/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > a > c > d
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 1/24 3/24 
agent 4: 0/24 0/24 21/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > c > a
agent 4: c > d > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > c > a
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 */24 */24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 */24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 4 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 */24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > c > a > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 4/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 */24 */24 0/24 
agent 2: 12/24 12/24 */24 */24 
agent 3: 0/24 */24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 */24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 */24 */24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 */24 0/24 
agent 2: 12/24 12/24 */24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 */24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: c > b > d > a
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: c > b > a > d
agent 4: d > b > c > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 */24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 4/24 0/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > a > c > b
The allocation takes the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: c > a > b > d
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Assume agent 2 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > c > b
agent 3: c > b > a > d
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 */24 */24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 2/24 4/24 
agent 3: 0/24 2/24 22/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > b > a > d
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 2/24 22/24 0/24 
agent 3: 4/24 6/24 2/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Assume agent 2 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > b > a > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > b > d > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: 8/24 */24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 8/24 0/24 
agent 2: 8/24 8/24 6/24 2/24 
agent 3: 8/24 8/24 6/24 2/24 
agent 4: 0/24 0/24 4/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: a > c > b > d
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 */24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 3/24 1/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 1/24 23/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > c > b > d
agent 3: b > a > d > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 8/24 0/24 
agent 2: 12/24 0/24 12/24 0/24 
agent 3: 0/24 20/24 3/24 1/24 
agent 4: 0/24 0/24 1/24 23/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > b > d > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > b > c > d
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 */24 */24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 8/24 2/24 6/24 
agent 2: 8/24 8/24 2/24 6/24 
agent 3: 8/24 8/24 2/24 6/24 
agent 4: 0/24 0/24 18/24 6/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: c > d > b > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: */24 */24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 2/24 6/24 0/24 16/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > b > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 0/24 8/24 4/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: d > a > c > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 11/24 10/24 */24 0/24 
agent 2: 11/24 2/24 11/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 11/24 10/24 3/24 0/24 
agent 2: 11/24 2/24 11/24 0/24 
agent 3: 2/24 3/24 7/24 12/24 
agent 4: 0/24 9/24 3/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > d > a
agent 3: d > a > c > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 20/24 1/24 3/24 0/24 
agent 2: 0/24 19/24 5/24 0/24 
agent 3: 4/24 0/24 8/24 12/24 
agent 4: 0/24 4/24 8/24 12/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > a > b > d
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 4/24 */24 */24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: b > d > a > c
agent 4: c > b > a > d
The allocation takes the form
         a   b   c   d
agent 1: 12/24 4/24 2/24 6/24 
agent 2: 12/24 0/24 0/24 12/24 
agent 3: 0/24 20/24 0/24 4/24 
agent 4: 0/24 0/24 22/24 2/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > b > a > d
agent 4: c > d > b > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 */24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 1/24 20/24 0/24 3/24 
agent 3: 2/24 4/24 12/24 6/24 
agent 4: 0/24 0/24 12/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 1 has flipped objects b and c.
Hence, at the profile
agent 1: a > b > c > d
agent 2: b > c > a > d
agent 3: b > d > c > a
agent 4: c > a > b > d
The allocation takes the form
         a   b   c   d
agent 1: 21/24 0/24 0/24 3/24 
agent 2: 2/24 12/24 4/24 6/24 
agent 3: 0/24 12/24 0/24 12/24 
agent 4: 1/24 0/24 20/24 3/24 
Note that we have also applied Neutrality to rename the objects to keep the preferences of the first agent unchanged. 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 3/24 20/24 1/24 
agent 4: 0/24 3/24 0/24 21/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 4/24 6/24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > a > d > b
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 4/24 6/24 12/24 */24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 4/24 6/24 12/24 2/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > b > c > a
We already know that the allocation has the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 */24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 0/24 12/24 0/24 12/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > a > b > d
agent 4: d > b > c > a
The allocation takes the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > c > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 4/24 */24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 6/24 4/24 2/24 
agent 3: 0/24 4/24 20/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > a > d > b
agent 3: c > b > d > a
agent 4: d > b > a > c
The allocation takes the form
         a   b   c   d
agent 1: 20/24 4/24 0/24 0/24 
agent 2: 4/24 6/24 12/24 2/24 
agent 3: 0/24 12/24 12/24 0/24 
agent 4: 0/24 2/24 0/24 22/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > d > c > a
agent 3: b > d > c > a
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 */24 4/24 
agent 3: 0/24 12/24 8/24 */24 
agent 4: 0/24 0/24 8/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 12/24 8/24 4/24 
agent 3: 0/24 12/24 8/24 4/24 
agent 4: 0/24 0/24 8/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: c > d > b > a
agent 4: d > b > a > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 */24 12/24 4/24 
agent 3: 0/24 8/24 12/24 */24 
agent 4: 0/24 8/24 0/24 16/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 8/24 12/24 4/24 
agent 3: 0/24 8/24 12/24 4/24 
agent 4: 0/24 8/24 0/24 16/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > d > b > c
agent 3: c > d > b > a
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 */24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 8/24 0/24 4/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 4/24 0/24 20/24 

Let's apply Strategy-Proofness.
Assume agent 2 has flipped objects a and b.
Hence, at the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > b > c
The allocation takes the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: 4/24 */24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > b > a
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: 4/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 8/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > d > a > b
agent 3: d > c > a > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: */24 */24 4/24 12/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 18/24 6/24 0/24 0/24 
agent 2: 2/24 6/24 16/24 0/24 
agent 3: 2/24 6/24 4/24 12/24 
agent 4: 2/24 6/24 4/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > d > c
agent 3: c > b > a > d
agent 4: d > b > a > c
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 12/24 12/24 0/24 0/24 
agent 2: 12/24 12/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > d > b
agent 4: d > c > a > b
We already know that the allocation has the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: */24 0/24 */24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Anonymity & No-waste Constraint gives 
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 

Let's apply Strategy-Proofness.
Assume agent 3 has flipped objects c and d.
Hence, at the profile
agent 1: a > b > c > d
agent 2: a > b > c > d
agent 3: a > c > b > d
agent 4: d > c > a > b
The allocation takes the form
         a   b   c   d
agent 1: 8/24 12/24 4/24 0/24 
agent 2: 8/24 12/24 4/24 0/24 
agent 3: 8/24 0/24 16/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > c > d
agent 3: c > a > d > b
agent 4: d > a > c > b
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: */24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


Consider the profile
agent 1: a > b > c > d
agent 2: c > a > b > d
agent 3: d > a > b > c
agent 4: d > a > b > c
We already know that the allocation has the form
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: */24 8/24 0/24 12/24 
agent 4: 4/24 */24 0/24 12/24 
Application of Lottery Constraint gives 
         a   b   c   d
agent 1: 16/24 8/24 0/24 0/24 
agent 2: 0/24 0/24 24/24 0/24 
agent 3: 4/24 8/24 0/24 12/24 
agent 4: 4/24 8/24 0/24 12/24 


Consider the profile
agent 1: a > b > c > d
agent 2: b > a > d > c
agent 3: c > d > b > a
agent 4: d > c > b > a
Application of Ex-post Efficiency gives 
         a   b   c   d
agent 1: 24/24 0/24 0/24 0/24 
agent 2: 0/24 24/24 0/24 0/24 
agent 3: 0/24 0/24 24/24 0/24 
agent 4: 0/24 0/24 0/24 24/24 


For all the preferences profiles the outcome of the mechanism was deduced from the axioms.
The conjecture is proved for N=4.
Hurah!!